Abstract
We review the intimate connection between (super)gravity close to a spacelike singularity (the “BKLlimit”) and the theory of Lorentzian KacMoody algebras. We show that in this limit the gravitational theory can be reformulated in terms of billiard motion in a region of hyperbolic space, revealing that the dynamics is completely determined by a (possibly infinite) sequence of reflections, which are elements of a Lorentzian Coxeter group. Such Coxeter groups are the Weyl groups of infinitedimensional KacMoody algebras, suggesting that these algebras yield symmetries of gravitational theories. Our presentation is aimed to be a selfcontained and comprehensive treatment of the subject, with all the relevant mathematical background material introduced and explained in detail. We also review attempts at making the infinitedimensional symmetries manifest, through the construction of a geodesic sigma model based on a Lorentzian KacMoody algebra. An explicit example is provided for the case of the hyperbolic algebra E_{10}, which is conjectured to be an underlying symmetry of Mtheory. Illustrations of this conjecture are also discussed in the context of cosmological solutions to elevendimensional supergravity.
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1 Introduction
It has been realized long ago that spacetime singularities are generic in classical general relativity [91]. However, their exact nature is still far from being well understood. Although it is expected that spacetime singularities will ultimately be resolved in a complete quantum theory of gravity, understanding their classical structure is likely to shed interesting light and insight into the nature of the mechanisms at play in the singularity resolution. Furthermore, analyzing general relativity close to such singularities also provides important information on the dynamics of gravity within the regime where it breaks down. Indeed, careful investigations of the field equations in this extreme regime has revealed interesting and unexpected symmetry properties of gravity.
In the late 1960’s, Belinskii, Khalatnikov and Lifshitz (“BKL”) [16] gave a general description of spacelike singularities in the context of the fourdimensional vacuum Einstein theory. They provided convincing evidence that the generic solution of the dynamical Einstein equations, in the vicinity of a spacelike singularity, exhibits the following remarkable properties:
The spatial points dynamically decouple, i.e., the partial differential equations governing the dynamics of the spatial metric asymptotically reduce, as one goes to the singularity, to ordinary differential equations with respect to time (one set of ordinary differential equations per spatial point).
The solution exhibits strong chaotic properties of the type investigated independently by Misner [137] and called “mixmaster behavior”. This chaotic behavior is best seen in the hyperbolic billiard reformulation of the dynamics due to Chitre [31] and Misner [138] (for pure gravity in four spacetime dimensions).
1.1 Cosmological billiards and hidden symmetries of gravity
This important work has opened the way to many further fruitful investigations in theoretical cosmology. Recently, a new — and somewhat unanticipated — development has occurred in the field, with the realisation that for the gravitational theories that have been studied most (pure gravity and supergravities in various spacetime dimensions) the dynamics of the gravitational field exhibits strong connections with Lorentzian KacMoody algebras, as discovered by Damour and Henneaux [45], suggesting that these might be “hidden” symmetries of the theory.
These connections appear for the cases at hand because in the BKLlimit, not only can the equations of motion be reformulated as dynamical equations for billiard motion in a region of hyperbolic space, but also this region possesses unique features: It is the fundamental Weyl chamber of some KacMoody algebra. The dynamical motion in the BKLlimit is then a succession of reflections in the walls bounding the fundamental Weyl chamber and defines “words” in the Weyl group of the KacMoody algebra.
Which billiard region of hyperbolic space actually emerges — and hence which KacMoody algebra is relevant — depends on the theory at hand, i.e., on the spacetime dimension, the menu of matter fields, and the dilaton couplings. The most celebrated case is elevendimensional supergravity, for which the billiard region is the fundamental region of \({E_{10}} \equiv E_8^{+ +}\), one of the four hyperbolic KacMoody algebras of highest rank 10. The root lattice of E_{10} is furthermore one of the few even, Lorentzian, selfdual lattices — actually the only one in 10 dimensions — a fact that could play a key role in our ultimate understanding of Mtheory.
Other gravitational theories lead to other billiards characterized by different algebras. These algebras are closely connected to the hidden duality groups that emerge upon dimensional reduction to three dimensions [41, 95].
That one can associate a regular billiard and an infinite discrete reflection group (Coxeter group) to spacelike singularities of a given gravitational theory in the BKLlimit is a robust fact (even though the BKLlimit itself is yet to be fully understood), which, in our opinion, will survive future developments. The mathematics necessary to appreciate the billiard structure and its connection to the duality groups in three dimensions involve hyperbolic Coxeter groups, KacMoody algebras and real forms of Lie algebras.
The appearance of infinite Coxeter groups related to Lorentzian KacMoody algebras has triggered fascinating conjectures on the existence of huge symmetry structures underlying gravity [47]. Similar conjectures based on different considerations had been made earlier in the pioneering works [113, 167]. The status of these conjectures, however, is still somewhat unclear since, in particular, it is not known how exactly the symmetry would act.
The main purpose of this article is to explain the emergence of infinite discrete reflection groups in gravity in a selfcontained manner, including giving the detailed mathematical background needed to follow the discussion. We shall avoid, however, duplicating already existing reviews on BKL billiards.
Contrary to the main core of the review, devoted to an explanation of the billiard Weyl groups, which is indeed rather complete, we shall also discuss some paths that have been taken towards revealing the conjectured infinitedimensional KacMoody symmetry. Our goal here will only be to give a flavor of some of the work that has been done along these lines, emphasizing its dynamical relevance. Because we feel that it would be premature to fully review this second subject, which is still in its infancy, we shall neither try to be exhaustive nor give detailed treatments.
1.2 Outline of the paper
Our article is organized as follows. In Section 2, we outline the key features of the BKL phenomenon, valid in any number of dimensions, and describe the billiard formulation which clearly displays these features. Since the derivation of these aspects have been already reviewed in [48], we give here only the results without proof. Next, for completeness, we briefly discuss the status of the BKL conjecture — assumed to be valid throughout our review.
In Sections 3 and 4, we develop the mathematical tools necessary for apprehending those aspects of Coxeter groups and KacMoody algebras that are needed in the BKL analysis. First, in Section 3, we provide a primer on Coxeter groups (which are the mathematical structures that make direct contact with the BKL billiards). We then move on to KacMoody algebras in Section 4, and we discuss, in particular, some prominent features of hyperbolic KacMoody algebras.
In Section 5 we then make use of these mathematical concepts to relate the BKL billiards to Lorentzian KacMoody algebras. We show that there is a simple connection between the relevant KacMoody algebra and the Uduality algebras that appear upon toroidal dimensional reduction to three dimensions, when these Uduality algebras are split real forms. The KacMoody algebra is then just the standard overextension of the Uduality algebra in question.
To understand the nonsplit case requires an understanding of real forms of finitedimensional semisimple Lie algebras. This mathematical material is developed in Section 6. Here, again, we have tried to be both rather complete and explicit through the use of many examples. We have followed a pedagogical approach privileging illustrative examples over complete proofs (these can be found in any case in the references given in the text). We explain the complementary Vogan and TitsSatake approaches, where maximal compact and maximal noncompact Cartan subalgebras play the central roles, respectively. The concepts of restricted root systems and of the Iwasawa decomposition, central for understanding the emergence of the billiard, have been given particular attention. For completeness we provide tables listing all real forms of finite Lie algebras, both in terms of Vogan diagrams and in terms of TitsSatake diagrams. In Section 7 we use these mathematical developments to relate the KacMoody billiards in the nonsplit case to the Uduality algebras appearing in three dimensions.
Up to (and including) Section 7, the developments present wellestablished results. With Section 8 we initiate a journey into more speculative territory. The presence of hyperbolic Weyl groups suggests that the corresponding infinitedimensional KacMoody algebras might, in fact, be true underlying symmetries of the theory. How this conjectured symmetry should actually act on the physical fields is still unclear, however. We explore one approach in which the symmetry is realized nonlinearly on a (1 + 0)dimensional sigma model based on \({{\mathcal E}_{10}}/{\mathcal K}({{\mathcal E}_{10}})\), which is the case relevant to elevendimensional supergravity. To this end, in Section 8 we introduce the concept of a level decomposition of some of the relevant KacMoody algebras in terms of finite regular subalgebras. This is necessary for studying the sigma model approach to the conjectured infinitedimensional symmetries, a task undertaken in Section 9. We show that the sigma model for \({{\mathcal E}_{10}}/{\mathcal K}({{\mathcal E}_{10}})\) spectacularly reproduces important features of elevendimensional supergravity. However, we also point out important limitations of the approach, which probably does not constitute the final word on the subject.
In Section 10 we show that the interpretation of elevendimensional supergravity in terms of a manifestly \({{\mathcal E}_{10}}\)invariant sigma model sheds interesting and useful light on certain cosmological solutions of the theory. These solutions were derived previously but without the KacMoody algebraic understanding. The sigma model approach also suggests a new method of uncovering novel solutions. Finally, in Section 11 we present a concluding discussion and some suggestions for future research.
2 The BKL Phenomenon
In this section, we explain the main ideas of the billiard description of the BKL behavior. Our approach is based on the billiard review [48], from which we adopt notations and conventions. We shall here only outline the logic and provide the final results. No attempt will be made to reproduce the (sometimes heuristic) arguments underlying the derivation.
2.1 The general action
We are interested in general theories describing Einstein gravity coupled to bosonic “matter” fields. The only known bosonic matter fields that consistently couple to gravity are pform fields, so our collection of fields will contain, besides the metric, pform fields, including scalar fields (p = 0). The action reads
where we have chosen units such that 16πG = 1. The spacetime dimension is left unspecified. The Einstein metric g_{μν} has Lorentzian signature (−, +, …, +) and is used to lower or raise the indices. Its determinant is^{(D)}g, where the index D is used to avoid any confusion with the determinant of the spatial metric introduced below. We assume that among the scalars, there is only one dilaton^{Footnote 1}, denoted ϕ, whose kinetic term is normalized with weight 1 with respect to the Ricci scalar. The real parameter λ^{(p)} measures the strength of the coupling to the dilaton. The other scalar fields, sometimes called axions, are denoted A^{(0)} and have dilaton coupling λ^{(0)} ≠ 0. The integer p ≥ 0 labels the various pforms A^{(p)} present in the theory, with field strengths F^{(p)} = dA^{(p)},
We assume the form degree p to be strictly smaller than D − 1, since a (D − 1)form in D dimensions carries no local degree of freedom. Furthermore, if p = D − 2 the pform is dual to a scalar and we impose also λ^{(D−2)} ≠ 0.
The field strength, Equation (2.2), could be modified by additional coupling terms of YangMills or ChaplineManton type [20, 29] (e.g., F_{C} = dC^{(2)} − C^{(0)}dB^{(2)} for two 2forms C^{(2)} and B^{(2)} and a 0form C^{(0)}, as it occurs in tendimensional type IIB supergravity), but we include these additional contributions to the action in “more”. Similarly, “more” might contain ChernSimons terms, as in the action for elevendimensional supergravity [38].
We shall at this stage consider arbitrary dilaton couplings and menus of pforms. The billiard derivation given below remains valid no matter what these are; all theories described by the general action Equation (2.1) lead to the billiard picture. However, it is only for particular pform menus, spacetime dimensions and dilaton couplings that the billiard region is regular and associated with a KacMoody algebra. This will be discussed in Section 5. Note that the action, Equation (2.1), contains as particular cases the bosonic sectors of all known supergravity theories.
2.2 Hamiltonian description
We assume that there is a spacelike singularity at a finite distance in proper time. We adopt a spacetime slicing adapted to the singularity, which “occurs” on a slice of constant time. We build the slicing from the singularity by taking pseudoGaussian coordinates defined by \(N = \sqrt g\) and N^{i} = 0, where N is the lapse and N^{i} is the shift [48]. Here, g = det(g_{ij}). Thus, in some spacetime patch, the metric reads^{Footnote 2}
where the local volume g collapses at each spatial point as x^{0} → +∞, in such a way that the proper time \(dT =  \sqrt g d{x^0}\) remains finite (and tends conventionally to 0+). Here we have assumed the singularity to occur in the past, as in the original BKL analysis, but a similar discussion holds for future spacelike singularities.
2.2.1 Action in canonical form
In the Hamiltonian description of the dynamics, the canonical variables are the spatial metric components g_{ij}, the dilaton ϕ, the spatial pform components \(A_{{m_1}\, \cdots {m_p}}^{(p)}\) and their respective conjugate momenta π^{ij}, πϕ and\(\pi _{(p)}^{{m_1}\, \cdots {m_p}}\). The Hamiltonian action in the pseudoGaussian gauge is given by
where the Hamiltonian is
In addition to imposing the coordinate conditions \(N = \sqrt g\) and N^{i} = 0, we have also set the temporal components of the pforms equal to zero (“temporal gauge”).
The dynamical equations of motion are obtained by varying the above action w.r.t. the canonical variables. Moreover, there are constraints on the dynamical variables, which are
Here we have set
where the subscript m_{p} denotes the spatially covariant derivative. These constraints are preserved by the dynamical evolution and need to be imposed only at one “initial” time, say at x^{0} = 0.
2.2.2 Iwasawa change of variables
In order to study the dynamical behavior of the fields as x^{0} → ∞ (g → 0) and to exhibit the billiard picture, it is particularly convenient to perform the Iwasawa decomposition of the spatial metric. Let g(x^{0},x^{i}) be the matrix with entries g_{ij}(x^{0},x^{i}). We set
where \({\mathcal N}\) is an upper triangular matrix with 1’s on the diagonal (\({{\mathcal N}_{ii}} = 1,\,{{\mathcal N}_{ij}} = 0\,{\rm{for}} \, i{\rm{>}}j\)) and A is a diagonal matrix with positive elements, which we parametrize as
Both \(\mathcal N\) and \(\mathcal A\) depend on the spacetime coordinates. The spatial metric dσ^{2} becomes
with
The variables β^{i} of the Iwasawa decomposition give the (logarithmic) scale factors in the new, orthogonal, basis. The variables \({{\mathcal N}_{ij}}\) characterize the change of basis that diagonalizes the metric and hence they parametrize the offdiagonal components of the original g_{ij}.
We extend the transformation Equation (2.8) in configuration space to a canonical transformation in phase space through the formula
Since the scale factors and the offdiagonal variables play very distinct roles in the asymptotic behavior, we split off the Hamiltonian as a sum of a kinetic term for the scale factors (including the dilaton),
plus the rest, denoted by V, which will act as a potential for the scale factors. The Hamiltonian then becomes
The kinetic term K is quadratic in the momenta conjugate to the scale factors and defines the inverse of a metric in the space of the scale factors. Explicitly, this metric reads
Since the metric coefficients do not depend on the scale factors, that metric in the space of scale factors is flat, and, moreover, it is of Lorentzian signature. A conformal transformation where all scale factors are scaled by the same number (β^{i} → β^{i} + ϵ) defines a timelike direction. It will be convenient in the following to collectively denote all the scale factors (the β^{i}’s and the dilaton ϕ) as β^{μ}, i.e., (β^{μ}) = (β^{i}ϕ).
The analysis is further simplified if we take for new pform variables the components of the pforms in the Iwasawa basis of the ω_{k},
and again extend this configuration space transformation to a point canonical transformation in phase space,
using the formula ∑ p dq = ∑p′ dq′, which reads
Note that the scale factor variables are unaffected, while the momenta P_{ij} conjugate to \({{\mathcal N}_{ij}}\) get redefined by terms involving \({\mathcal E}\), \({\mathcal N}\) and \({\mathcal A}\) since the components \(A_{{m_1}\, \cdots {m_p}}^{(p)}\) of the pforms in the Iwasawa basis involve the \({\mathcal N}\)’s. On the other hand, the new pform momenta, i.e., the components of the electric field \(\pi _{(p)}^{{m_{1\, \cdots}}{m_p}}\) in the basis {ω^{k}} are simply given by
In terms of the new variables, the electromagnetic potentials become
Here, \({e_{{i_1} \ldots {i_p}}}(\beta)\) are the electric linear forms
(the indices i_{j} are all distinct because \({\mathcal E}_{(p)}^{{i_1} \cdots {i_p}}\) is completely antisymmetric) while \({{\mathcal F}_{(p)\,{i_1} \cdots {i_{p + 1}}}}\) are the components of the magnetic field \({F_{(p){m_1}\, \cdots {m_{p + 1}}}}\) in the basis {ω^{k}},
and \({m_{{i_1} \cdots {i_{p + 1}}}}(\beta)\) are the magnetic linear forms
One sometimes rewrites \({m_{{i_1} \cdots {i_{p + 1}}}}(\beta)\) as \({b_{{i_{p + 2 \cdots {i_d}}}}}(\beta)\), where {i_{p+2}, i_{p+3}, …, i_{d}} is the set complementary to {i_{1}, i_{2}, … i_{p+1}}, e.g.,
The exterior derivative \({\mathcal F}\) of \({\mathcal A}\) in the nonholonomic frame {ω^{k}} involves of course the structure coefficients \({C^i}_{jk}\) in that frame, i.e.,
where
is here the frame derivative. Similarly, the potential V_{ϕ} reads
where \({{\mathcal F}_i}\) is
and
2.3 Decoupling of spatial points close to a spacelike singularity
So far we have only redefined the variables without making any approximation. We now start the discussion of the BKLlimit, which investigates the leading behavior of the fields as x^{0} → ∞ (g → 0). Although the more recent “derivations” of the BKLlimit treat both elements at once [43, 44, 45, 48], it appears useful — especially for rigorous justifications — to separate two aspects of the BKL conjecture^{Footnote 3}.
The first aspect is that the spatial points decouple in the limit x^{0} → ∞, in the sense that one can replace the Hamiltonian by an effective “ultralocal” Hamiltonian H^{UL} involving no spatial gradients and hence leading at each point to a set of dynamical equations that are ordinary differential equations with respect to time. The ultralocal effective Hamiltonian has a form similar to that of the Hamiltonian governing certain spatially homogeneous cosmological models, as we shall explain in this section.
The second aspect of the BKLlimit is to take the sharp wall limit of the ultralocal Hamiltonian. This leads directly to the billiard description, as will be discussed in Section 2.4.
2.3.1 Spatially homogeneous models
In spatially homogeneous models, the fields depend only on time in invariant frames, e.g., for the metric
where the invariant forms fulfill
Here, the \({f^i}_{jk}\) are the structure constants of the spatial homogeneity group. Similarly, for a 1form and a 2form,
The Hamiltonian constraint yielding the field equations in the spatially homogeneous context^{Footnote 4} is obtained by substituting the form of the fields in the general Hamiltonian constraint and contains, of course, no explicit spatial gradients since the fields are homogeneous. Note, however, that the structure constants \({f^i}_{jk}\) contain implicit spatial gradients. The Hamiltonian can now be decomposed as before and reads
where K, V_{S} and \(V_{(p)}^{{\rm{el}}}\) which do not involve spatial gradients, are unchanged and where V_{ϕ} disappears since \({\partial _i}\phi = 0\). The potential V_{G} is given by [61]
where the linear forms α_{ijk}(β) (with i, j, k distinct) read
and where “more” stands for the terms in the first sum that arise upon taking i = j or i = k. The structure constants in the Iwasawa frame (with respect to the coframe in Equation (2.30)) are related to the structure constants \({f^i}_{jk}\) through
and depend therefore on the dynamical variables. Similarly, the potential \(V_{(p)}^{{\rm{magn}}}\) becomes
where the field strengths \({\mathcal F}_{(p)\,{i_1} \cdots {i_{p + 1}}}^h\) reduce to the “AC” terms in dA and depend on the potentials and the offdiagonal Iwasawa variables.
2.3.2 The ultralocal Hamiltonian
Let us now come back to the general, inhomogeneous case and express the dynamics in the frame {dx^{0}, ψ^{i}} where the ψ^{i}’s form a “generic” nonholonomic frame in space,
Here the \({f^i}_{jk}\)’s are in general spacedependent. In the nonholonomic frame, the exact Hamiltonian takes the form
where the ultralocal part \({{\mathcal H}^{{\rm{UL}}}}\) is given by Equations (2.32) and (2.33) with the relevant \({f^i}_{jk}\)’s, and where \({{\mathcal H}^{{\rm{gradient}}}}\) involves the spatial gradients of \({f^i}_{jk},\,{\beta ^m},\,\phi\) and \({{\mathcal N}_{ij}}\).
The first part of the BKL conjecture states that one can drop \({{\mathcal H}^{{\rm{gradient}}}}\) asymptotically; namely, the dynamics of a generic solution of the Einsteinpformdilaton equations (not necessarily spatially homogeneous) is asymptotically determined, as one goes to the spatial singularity, by the ultralocal Hamiltonian
provided that the phase space constants \({f^i}_{jk}({x^m}) =  {f^i}_{kj}({x^m})\) are such that all exponentials in the above potentials do appear. In other words, the f’s must be chosen such that none of the coefficients of the exponentials, which involve f and the fields, identically vanishes — as would be the case, for example, if \({f^i}_{jk} = 0\) since then the potentials V_{G} and \(V_{(p)}^{{\rm{magn}}}\) are equal to zero. This is always possible because the \({f^i}_{jk}\), even though independent of the dynamical variables, may in fact depend on x and so are not required to fulfill relations “f f = 0” analogous to the Bianchi identity since one has instead “∂f + f f = 0”.
2.3.2.1 Comments

1.
As we shall see, the conditions on the f’s (that all exponentials in the potential should be present) can be considerably weakened. It is necessary that only the relevant exponentials (in the sense defined in Section 2.4) be present. Thus, one can correctly capture the asymptotic BKL behavior of a generic solution with fewer exponentials. In the case of elevendimensional supergravity the spatial curvature is asymptotically negligible with respect to the electromagnetic terms and one can in fact take a holonomic frame for which \({f^i}_{jk} = 0\) (and hence also \({C^i}_{jk} = 0\).

2.
The actual values of the \({f^i}_{jk}\) (provided they fulfill the criterion given above or rather its weaker form just mentioned) turn out to be irrelevant in the BKLlimit because they can be absorbed through redefinitions. This is for instance why the Bianchi VIII and IX models, even though they correspond to different groups, can both be used to describe the BKL behavior in four spacetime dimensions.
2.4 Dynamics as a billiard in hyperbolic space
The second step in the BKLlimit is to take the sharp wall limit of the potentials.^{Footnote 5} This leads to the billiard picture. It is crucial here that the coefficients in front of the dominant walls are all positive. Again, just as for the first step, this limit has not been fully justified. Only heuristic, albeit convincing, arguments have been put forward.
The idea is that as one goes to the singularity, the exponential potentials get sharper and sharper and can be replaced in the limit by the corresponding Θ∞function, denoted for short Θ and defined by Θ(x) = 0 for x < 0 and Θ(x) = +∞ for x > 0. Taking into account the facts that aΘ(x) = Θ(x) for all a > 0, as well as that some walls can be neglected, one finds that the Hamiltonian becomes in the sharp wall limit
with
where s_{ji}(β) = β^{j} − β^{i}. See [48] for more information.
The description of the motion of the scale factors (at each spatial point) is easy to give in that limit. Because the potential walls are infinite (and positive), the motion is constrained to the region where the arguments of all Θfunctions are negative, i.e., to
In that region, the motion is governed by the kinetic term K, i.e., is a geodesic for the metric in the space of the scale factors. Since that metric is flat, this is a straight line. In addition, the constraint \({\mathcal H} = 0\), which reduces to K=0 away from the potential walls, forces the straight line to be null. We shall assume that the time orientation in the space of the scale factors is such that the straight line is futureoriented (g → 0 in the future).
It is easy to check that all the walls appearing in Equation (2.41), collectively denoted \({F_A}(\beta) \equiv {F_{A\mu}}{\beta ^\mu} = 0\), are timelike hyperplanes. This is because the squared norms of all the F_{a}’s are positive,
Explicitly, one finds
Because the potential walls are timelike, they have a nonempty intersection with the forward light cone in the space of the scale factors. When the null straight line representing the evolution of the scale factors hits one of the walls, it gets reflected according to the rule [43]
where v is the velocity vector (tangent to the straight line). This reflection preserves the time orientation since the hyperplanes are timelike and hence belong to the orthochronous Lorentz group O^{↑} (k, 1) where k = d − 1 or d according to whether there is no or one dilaton. The conditions s_{ji} = 0 define the “symmetry” or “centrifugal” walls, the conditions β_{ijk} = 0 define the “curvature” or “gravitational” walls, the conditions \({\alpha _{ijk}} = 0\) define the “electric” walls, while the conditions \({m_{{i_1} \cdots {i_{p + 1}}}} = 0\) define the “magnetic” walls.
The motion is thus a succession of futureoriented null straight line segments interrupted by reflections against the walls, where the motion undergoes a reflection belonging to O^{↑}(k, 1). Whether the collisions eventually stop or continue forever is better visualized by projecting the motion radially on the positive sheet of the unit hyperboloid, as was done first in the pioneering work of Chitre and Misner [31, 138] for pure gravity in four spacetime dimensions. We recall that the positive sheet of the unit hyperboloid ∑ (β^{i})^{2} − β^{i})^{2} + ϕ^{2} = −1, ∑β^{i} > 0, provides a model of hyperbolic space (see, e.g., [146]).
The intersection of a timelike hyperplane with the unit hyperboloid defines a hyperplane in hyperbolic space. The region in hyperbolic space on the positive side of all hyperplanes is the allowed dynamical region and is called the “billiard table”. It is never compact in the cases relevant to gravity, but it may or may not have finite volume. The projection of the motion of the scale factors on the unit hyperboloid is the same as the motion of a billiard ball in a hyperbolic billiard: geodesic arcs in hyperbolic space within the billiard region, interrupted by collisions against the bounding walls where the motion undergoes a specular reflection.
When the volume of the billiard table is finite, the collisions with the potential walls never end (for generic initial data) and the motion is chaotic. When, on the other hand, the volume is infinite, generic initial data lead to a motion that ultimately freely runs away to infinity. This is nonchaotic. For more information, see [135, 170]. An interesting criterion for chaos (equivalent to finite volume of hyperbolic billiard region) has been given in [111] in terms of illuminations of spheres by point sources.
2.4.1 Comments

1.
The task of determining the billiard region is greatly simplified by the observation that some walls are behind others and are thus not relevant. For instance, it is clear that if β^{2} − β^{1} > 0 and β^{3} − β^{2} > 0, then β^{3} − β^{1} > 0. Among the symmetry wall conditions, the only relevant ones are \({\beta ^{i + 1}}  {\beta ^i} > 0, \, i = 1,\,2,\, \cdots, \, d  1\) − β^{i} > 0, i = 1. 2. …, d − 1. Similarly, a wall of any given type can be written as a positive combination of the walls of the same type with smallest values of the indices i of the β’s and the symmetry walls (e.g., the electric wall condition β^{2} > 0 for a 1form with zero dilaton coupling can be written as β^{1} + (β^{2} − β^{1}) > 0 and is thus a consequence of β^{1} > 0 and β^{2} − β^{1} > 0). Finally, one also verifies that in the presence of true pforms (0 < p < d − 1), the gravitational walls are never relevant as they can be written as combinations of pform walls with positive coefficients [49].

2.
It is interesting to determine the spatially homogeneous models that reproduce asymptotically the correct billiard limit. It is clear that in order to do so, homogeneous cosmological models need only contain the relevant walls. It is not necessary that they yield all the walls. Which homogeneity groups are acceptable depends on the system at hand. We list here a few examples. For vacuum gravity in four spacetime dimensions, the appropriate homogeneous models are the socalled Bianchi VIII or IX models. For vacuum gravity in higher dimensions, the structure constants of the homogeneity group must fulfill the conditions of [60] and the metric must include offdiagonal components (see also [58]). In the presence of a single pform and no dilaton (0 < p < d − 1), the simplest (Abelian) homogeneity group can be taken [44].
2.5 Rules for deriving the wall forms from the Lagrangian — Summary
We have recalled above that the generic behavior near a spacelike singularity of the system with action (2.1) can be described at each spatial point in terms of a billiard in hyperbolic space. The action for the billiard ball reads, in the gauge \(N = \sqrt g\),
where we recall that x^{0} → ∞ in the BKLlimit (proper time T → 0^{+}), and G_{μν} is the metric in the space of the scale factors,
introduced in Equation (2.15) above. As stressed there, this metric is flat and of Lorentzian signature. Between two collisions, the motion is a free, geodesic motion. The collisions with the walls are controlled by the potential V(β^{μ}), which is a sum of sharp wall potentials. The walls are hyperplanes and can be inferred from the Lagrangian. They are as follows:
 1.
Gravity brings in the symmetry walls
$${\beta ^{i + 1}}  {\beta ^i} = 0,$$(2.48)with i = 1, 2, …,d − 1, and the curvature wall
$$2{\beta ^1} + {\beta ^2} + \cdots + {\beta ^{d  2}} = 0.$$(2.49)  2.
Each pform brings in an electric wall
$${\beta ^1} + \cdots + {\beta ^p} + {{{\lambda ^{(p)}}} \over 2}\phi = 0,$$(2.50)and a magnetic wall
$${\beta ^1} + \cdots + {\beta ^{d  p  1}}  {{{\lambda ^{(p)}}} \over 2}\phi = 0.$$(2.51)
We have written here only the (potentially) relevant walls. There are other walls present in the potential, but because these are behind the relevant walls, which are infinitely steep in the BKLlimit, they are irrelevant. They are relevant, however, when trying to exhibit the symmetry in a complete treatment where the BKLlimit is the zeroth order term in a gradient expansion yet to be understood [47].
The scalar product dual to the scalar product in the space of the scale factors is
for two linear forms \(F = {F_i}{\beta ^i} + {F_\phi}\phi, \, G = {G_i}{\beta ^i} + {G_\phi}\phi\).
These recipes are all that we shall need for investigating the regularity properties of the billiards associated with the class of actions Equation (2.1).
2.6 More on the free motion: The Kasner solution
The free motion between two bounces is a straight line in the space of the scale factors. In terms of the original metric components, it takes the form of the Kasner solution with dilaton. Indeed, the free motion is given by
where the “velocities” q^{μ} are subject to
since the motion is lightlike by the Hamiltonian constraint. The proper time \(dT =  \sqrt g d{x^0}\) is then T = B exp(−Kx^{0}), with K = ∑_{i}q^{i} and for some constant B (we assume, as before, that the singularity is at T = 0^{+}). Redefining then
yields the celebrated Kasner solution
subject to the constraints
where A is a constant of integration and where the coordinates x^{i} have been suitably rescaled (if necessary).
2.7 Chaos and billiard volume
With our rules for writing down the billiard region, one can determine in which case the volume of the billiard is finite and in which case it is infinite. The finitevolume, chaotic case is also called “mixmaster case”, a terminology introduced in four dimensions in [137].
The following results have been obtained:
Pure gravity in D ≤ 10 dimensions is chaotic, but ceases to be so for D ≥ 11 [63, 62].
The introduction of a dilaton removes chaos [15, 3]. The gravitational fourderivative action in four dimensions, based on R^{2}, is dynamically equivalent to Einstein gravity coupled to a dilaton [160]. Hence, chaos is removed also for this case.
pform gauge fields (0 < p < d − 1) without scalar fields lead to a finitevolume billiard [44].
When both pforms and dilatons are included, the situation is more subtle as there is a competition between two opposing effects. One can show that if the dilaton couplings are in a “subcritical” open region that contains the origin — i.e., “not too big” — the billiard volume is infinite and the system is non chaotic. If the dilaton couplings are outside of that region, the billiard volume is finite and the system is chaotic [49].
2.8 A note on the constraints
We have focused in the above presentation on the dynamical equations of motion. The constraints were only briefly mentioned, with no discussion, except for the Hamiltonian constraint. This is legitimate because the constraints are first class and hence preserved by the Hamiltonian evolution. Thus, they need only be imposed at some “initial” time. Once this is done, one does not need to worry about them any more. Furthermore the momentum constraints and Gauss’ law constraints are differential equations relating the initial data at different spatial points. This means that they do not constrain the dynamical variables at a given point but involve also their gradients — contrary to the Hamiltonian constraint which becomes ultralocal. Consequently, at any given point, one can freely choose the initial data on the undifferentiated dynamical variables and then use these data as (part of) the appropriate boundary data necessary to integrate the constraints throughout space. This is why one can assert that all the walls described above are generically present even when the constraints are satisfied.
The situation is different in homogeneous cosmologies where the symmetry relates the values of the fields at all spatial points. The momentum and Gauss’ law constraints become then algebraic equations and might remove some relevant walls. But this feature (removal of walls by the momentum and Gauss’ law constraints) is specific to some homogeneous cosmologies and does not hold in the generic case where spatial gradients are nonzero.
A final comment: How the spatial diffeomorphism constraints and Gauss’ law fit in the conjectured infinitedimensional symmetry is a point that is still poorly understood. See, however, [52] for recent progress in this direction.
2.9 On the validity of the BKL conjecture — A status report
Providing a complete rigorous justification of the above description of the behavior of the gravitational field in the vicinity of a spacelike singularity is a formidable task that has not been pushed to completion yet. The task is formidable because the Einstein equations form a complicated nonlinear system of partial differential equations. We shall assume throughout our review that the BKL description is correct, based on the original convincing arguments put forward by BKL themselves [16] and the subsequent fruitful investigations that have shed further important light on the validity of the conjecture. The billiard description will thus be taken for granted.
For completeness, we provide in this section a short guide to the work that has been accumulated since the late 1960’s to consolidate the BKL phenomenon.
As we have indicated, there are two aspects to the BKL conjecture:
 1.
The first part of the conjecture states that spatial points decouple as one goes to a spacelike singularity in the sense that the evolution can be described by a collection of systems of ordinary differential equations with respect to time, one such system at each spatial point. (“A spacelike singularity is local.”)
 2.
The second part of the conjecture states that the system of ordinary differential equations with respect to time describing the asymptotic dynamics at any given spatial point can be asymptotically replaced by the billiard equations. If the matter content is such that the billiard table has infinite volume, the asymptotic behavior at each point is given by a (generalized) Kasner solution (“Kasnerlike spacelike singularities”). If, on the other hand, the matter content is such that the billiard table has finite volume, the asymptotic behavior at each point is a chaotic, infinite, oscillatory succession of Kasner epochs. (“Oscillatory, or mixmaster, spacelike singularities.”)
A third element of the original conjecture was that the matter could be neglected asymptotically. While generically true in four spacetime dimensions (the exception being a massless scalar field, equivalent to a fluid with the stiff equation of state p = ρ), this aspect of the conjecture does not remain valid in higher dimensions where the pform fields might add relevant walls that could change the qualitative asymptotic behavior. We shall thus focus here only on Aspects 1 and 2.
In the Kasnerlike case, the mathematical situation is easier to handle since the conjectured asymptotic behavior of the fields is then monotone and known in closed form. There exist theorems validating (generically) this conjectured asymptotic behavior, starting from the pioneering work of [3] (where the singularities with this behavior are called “quiescent”), which was extended later in [49] to cover more general matter contents. See also [18, 108] for related work.
The situation is much more complicated in the oscillatory case, where only partial results exist. However, even though as yet incomplete, the mathematical and numerical studies of the BKL analysis has provided overwhelming support for its validity. Most work has been done in four dimensions.
The first attempts to demonstrate that spacelike singularities are local were done in the simpler context of solutions with isometries. It is only recently that general solutions without symmetries have been treated, but this has been found to be possible only numerically so far [87]. The literature on this subject is vast and we refer to [2, 87, 147] for points of entry into it. Let us note that an important element in the analysis has been a more precise reformulation of what is meant by “local”. This has been achieved in [163], where a precise definition involving a judicious choice of scale invariant variables has been proposed and given the illustrative name of “asymptotic silence” — the singularities being called “silent singularities” since propagation of information is asymptotically eliminated.
If one accepts that generic spacelike singularities are silent, one can investigate the system of ordinary differential equations that arise in the local limit. In four dimensions, this system is the same as the system of ordinary differential equations describing the dynamics of spatially homogeneous cosmologies of Bianchi type IX. It has been effectively shown analytically in [151] that the Bianchi IX evolution equations can indeed be replaced, in the generic case, by the billiard equations (with only the dominant, sharp walls) that produce the mixmaster behavior. This validates the second element in the BKL conjecture in four dimensions.
The connection between the billiard variables and the scale invariant variables has been investigated recently in the interesting works [92, 162].
Finally, taking for granted the BKL conjecture, one might analyze the chaotic properties of the billiard map (when the volume is finite). Papers exploring this issue are [30, 32, 121, 132] (four dimensions) and [68] (five dimensions).
Let us finally mention the interesting recent paper [40], in which a more precise formulation of the BKL conjecture, aimed towards the chaotic case, is presented. In particular, the main result of this work is an extension of the Fuchsian techniques, employed, e.g., in [49], which are applicable also for systems exhibiting chaotic dynamics. Furthermore, [40] examines the geometric structure which is preserved close to the singularity, and it is shown that this structure has a mathematical description in terms of a so called “partially framed flag”.
3 Hyperbolic Coxeter Groups
In this section, we develop the theory of Coxeter groups with a particular emphasis on the hyperbolic case. The importance of Coxeter groups for the BKL analysis stems from the fact that in the case of the gravitational theories that have been studied most (pure gravity, supergravities), the group generated by the reflections in the billiard walls is a Coxeter group. This follows, in turn, from the regularity of the corresponding billiards, whose walls intersect at angles that are integer submultiples of π.
3.1 Preliminary example: The BKL billiard (vacuum D = 4 gravity)
To illustrate the regularity of the gravitational billiards and motivate the mathematical developments through an explicit example, we first compute in detail the billiard characterizing vacuum, D = 4 gravity. Since this corresponds to the case originally considered by BKL, we call it the “BKL billiard”. We show in detail that the billiard reflections in this case are governed by the “extended modular group” PGL(2. ℤ), which, as we shall see, is isomorphic to the hyperbolic Coxeter group \(A_1^{+ +}\).
3.1.1 Billiard reflections
There are three scale factors so that after radial projection on the unit hyperboloid, we get a billiard in twodimensional hyperbolic space. The billiard region is defined by the following relevant wall inequalities,
(symmetry walls) and
(curvature wall). The remarkable properties of this region from our point of view are:
It is a triangle (i.e., a simplex in two dimensions) because even though we had to begin with 6 walls (3 symmetry walls and 3 curvature walls), only 3 of them are relevant.
The walls intersect at angles that are integer submultiples of π, i.e., of the form
$${\pi \over n},$$(3.3)where n is an integer. The symmetry walls intersect indeed at sixty degrees (n = 3) since the scalar product of the corresponding linear forms (of norm squared equal to 2) is −1, while the gravitational wall makes angles of zero (n = ∞, scalar product = − 2) and ninety (n = 2, scalar product = 0) degrees with the symmetry walls.
These angles are captured in the matrix A = (A_{ij})_{i,j=1,2,3} of scalar products,
which reads explicitly
Recall from the previous section that the scalar product of two linear forms F = F_{i}β^{i} and \(G = {G_i}{\beta ^i}\) is, in a threedimensional scale factor space,
where we have taken α_{1}(β) = 2β^{1}, α_{2}(β) = β^{2} − β^{1} and α_{3}(β) = β^{3} − β^{2}. The corresponding billiard region is drawn in Figure 1.
Because the angles between the reflecting planes are integer submultiples of π, the reflections in the walls bounding the billiard region^{Footnote 6},
obey the following relations,
The product s_{1}s_{3} is a rotation by 2π/2 = π and hence squares to one; the product s_{2}s_{3} is a rotation by 2π/3 and hence its cube is equal to one. There is no power of the product s_{1}s_{2} that is equal to one, something that one conventionally writes as
The group generated by the reflections s_{1}, s_{2} and s_{3} is denoted \(A_1^{+ +}\), for reasons that will become clear in the following, and coincides with the arithmetic group PGL(2, ℤ), as we will now show (see also [75, 116, 107]).
3.1.2 On the group PGL(2, ℤ)
The group PGL(2. ℤ) is defined as the group of 2 × 2 matrices C with integer entries and determinant equal to ±1, with the identification of C and −C,
Note that although elements of the real general linear group GL(2, ℝ) have (nonvanishing) unrestricted determinants, the discrete subgroup GL(2. ℝ) ⊂ GL(2. ℝ) only allows for det C = ±1 in order for the inverse C^{−1} to also be an element of GL(2. ℤ).
There are two interesting realisations of PGL(2. ℤ) in terms of transformations in two dimensions:
One can view PGL(2. ℤ) as the group of fractional transformations of the complex plane
$$C:z \rightarrow z\prime = {{az + b} \over {cz + d}},\qquad a,b,c,d \in {\mathbb Z},$$(3.11)with
$$ad  cd = \pm 1.$$(3.12)Note that one gets the same transformation if C is replaced by −C, as one should. It is an easy exercise to verify that the action of PGL(2. ℤ) when defined in this way maps the complex upper halfplane,
$${\rm{{\mathbb H}}} = \{z \in {\rm{{\mathbb C}}}\,\vert \,\Im z > 0\} ,$$(3.13)onto itself whenever the determinant ad − bc of C is equal to +1. This is not the case, however, when det C = −1.
For this reason, it is convenient to consider alternatively the following action of PGL(2. ℤ),
$$\begin{array}{*{20}c} {z \rightarrow z\prime = {{az + b} \over {cz + d}},\quad \,\;\;\;\;\;\;{\rm{if}}\;ad  cb = 1,} \\ {{\rm{or}}} \\ {z \rightarrow z\prime = {{a\bar z + b} \over {c\bar z + d}},\quad \quad \,{\rm{if}}\;ad  cb =  1,} \\\end{array}$$(3.14)(a. b. c. d ∈ ℤ), which does map the complex upperhalf plane onto itself, i.e., which is such that \(\mathfrak I {z\prime} > 0\) whenever \(\mathfrak I z > 0\).
The transformation (3.14) is the composition of the identity with the transformation (3.11) when det C =1, and of the complex conjugation transformation, \(f:z \rightarrow \bar z\) with the transformation (3.11) when det C = −1. Because the coefficients a, b, c, and d are real, f commutes with C and furthermore the map (3.11) → (3.14) is a group isomorphism, so that we can indeed either view the group PGL(2, →) as the group of fractional transformations (3.11), or as the group of transformations (3.14).
An important subgroup of the group PGL(2, ℤ) is the group PSL(2, ℤ) for which ad − cb = 1, also called the “modular group”. The translation T: z → z +1 and the inversion S: z → −1/z are examples of modular transformations,
It is a classical result that any modular transformation can be written as the product
but the representation is not unique [4].
Let s_{1}, s_{2} and s_{3} be the PGL(2, ℤ)transformations
to which there correspond the matrices
The s_{i}’s are reflections in the straight lines x = 0, x =1/2 and the unit circle z = 1, respectively. These are in fact just the transformations of hyperbolic space s_{1}, s_{2} and s_{3} described in Section 3.1.1, since the reflection lines intersect at 0, 90 and 60 degrees, respectively.
One easily verifies that T = s_{2}s_{1} and that S = s_{1}s_{3} = s_{3}s_{1}. Since any transformation of PGL(2, ℤ) not in PSL(2, ℤ) can be written as a transformation of PSL(2, ℤ) times, say, s_{1} and since any transformation of PSL(2, ℤ) can be written as a product of S’s and T’s, it follows that the group generated by the 3 reflections s_{1}, s_{2} and s_{3} coincides with PGL(2, ℤ), as announced above. (Strictly speaking, PGL(2, ℤ) could be a quotient of that group by some invariant subgroup, but one may verify that the kernel of the homomorphism is trivial (see Section 3.2.5 below).) The fundamental domains for PGL(2, ℤ) and PSL(2, ℤ) are drawn in Figure 2. The equivalence between PGL(2, ℤ) and the Coxeter group \(A_1^{+ +}\) has been discussed previously in [75, 116, 107].
3.2 Coxeter groups — The general theory
We have just shown that the billiard group in the case of pure gravity in four spacetime dimensions is the group PGL(2, ℤ). This group is generated by reflections and is a particular example of a Coxeter group. Furthermore, as we shall explain below, this Coxeter group turns out to be the Weyl group of the (hyperbolic) KacMoody algebra \(A_1^{+ +}\). Our first encounter with Lorentzian KacMoody algebras in more general gravitational theories will also be through their Weyl groups, which are, exactly as in the fourdimensional case just described, particular instances of (nonEuclidean) Coxeter groups, and which arise as the groups of billiard reflections.
For this reason, we start by developing here some aspects of the theory of Coxeter groups. An excellent reference on the subject is [107], to which we refer for more details and information. We consider KacMoody algebras in Section 4.
3.2.1 Examples
Coxeter groups generalize the familiar notion of reflection groups in Euclidean space. Before we present the basic definition, let us briefly discuss some more illuminating examples.
3.2.1.1 The dihedral group I_{2}(3) = A_{2}
Consider the dihedral group I_{2}(3) of order 6 of symmetries of the equilateral triangle in the Euclidean plane.
This group contains the identity, three reflections s_{1}, s_{2} and s_{3} about the three medians, the rotation R_{1} of 2π/3 about the origin and the rotation R_{2} of 4π/3 about the origin (see Figure 3),
The reflections act as follows^{Footnote 7},
where () is here the Euclidean scalar product and where α_{i} is a vector orthogonal to the hyperplane (here, line) of reflection.
Now, all elements of the dihedral group I_{2}(3) can be written as products of the two reflections s_{1} and s_{2}:
Hence, the dihedral group I_{2}(3) is generated by s_{1} and s_{2}. The writing Equation (3.21) is not unique because s_{1} and s_{2} are subject to the following relations,
The first two relations merely follow from the fact that s_{1} and s_{2} are reflections, while the third relation is a consequence of the property that the product s_{1}s_{2} is a rotation by an angle of 2π/3. This follows, in turn, from the fact that the hyperplanes (lines) of reflection make an angle of π/3. There is no other relation between the generators s_{1} and s_{2} because any product of them can be reduced, using the relations Equation (3.22), to one of the 6 elements in Equation (3.21), and these are independent.
The dihedral group I_{2}(3) is also denoted A_{2} because it is the Weyl group of the simple Lie algebra A2 (see Section 4). It is isomorphic to the permutation group S3 of three objects.
3.2.1.2 The infinite dihedral group \({I_2}(\infty) \equiv A_1^ +\)
Consider now the group of isometries of the Euclidean line containing the symmetries about the points with integer or halfinteger values of x (x is a coordinate along the line) as well as the translations by an integer. This is clearly an infinite group. It is generated by the two reflections s1 about the origin and s_{2} about the point with coordinate 1/2,
The product s_{2}s_{1} is a translation by +1 while the product s_{1}s_{2} is a translation by −1, so no power of s_{1}s_{2} or s_{2}s_{1} gives the identity. All the powers (s_{2}s_{1})^{k} and (s_{1}s_{2})^{j} are distinct (translations by +k and −j, respectively). The only relations between the generators are
This infinite dihedral group I_{2}(∞) is also denoted by \(A_1^ +\) because it is the Weyl group of the affine KacMoody algebra \(A_1^ +\).
3.2.2 Definition
A Coxeter group \({\mathfrak C}\) is a group generated by a finite number of elements s_{i} (i = 1. …,n) subject to relations that take the form
and
where the integers m_{ij} associated with the pairs (i, j) fulfill
Note that Equation (3.25) is a particular case of Equation (3.26) with m_{ii} = 1. If there is no power of s_{i}s_{j} that gives the identity, as in our second example, we set, by convention, m_{ij} = ∞. The generators s_{i} are called “reflections” because of Equation (3.25), even though we have not developed yet a geometric realisation of the group. This will be done in Section 3.2.4 below.
The number n of generators is called the rank of the Coxeter group. The Coxeter group is completely specified by the integers m_{ij}. It is useful to draw the set {m_{ij}} pictorially in a diagram Γ, called a Coxeter graph. With each reflection s_{i}, one associates a node. Thus there are n nodes in the diagram. If m_{ij} > 2, one draws a line between the node i and the node j and writes m_{ij} over the line, except if m_{ij} is equal to 3, in which case one writes nothing. The default value is thus “3”. When there is no line between i and j (i ≠ j), the exponent m_{ij} is equal to 2. We have drawn the Coxeter graphs for the Coxeter groups I_{2}(3), I_{2}(m) and for the Coxeter group H_{3} of symmetries of the icosahedron.
Note that if m_{ij} = 2, the generators s_{i} and s_{j} commute, s_{i}s_{j} = s_{i}s_{j}. Thus, a Coxeter group \({\mathfrak C}\) is the direct product of the Coxeter subgroups associated with the connected components of its Coxeter graph. For that reason, we can restrict the analysis to Coxeter groups associated with connected (also called irreducible) Coxeter graphs.
The Coxeter group may be finite or infinite as the previous examples show.
3.2.2.1 Another example: \(C_2^ +\)
It should be stressed that the Coxeter group can be infinite even if none of the Coxeter exponent is infinite. Consider for instance the group of isometries of the Euclidean plane generated by reflections in the following three straight lines: (i) the xaxis (s_{1}), (ii) the straight line joining the points (1,0) and (0,1) (s_{2}), and (iii) the yaxis (s_{3}). The Coxeter exponents are finite and equal to 4 (m_{12} = m_{21} = m_{23} = m_{32} = 4) and 2 (m_{13} = m_{31} = 2). The Coxeter graph is given in Figure 7. The Coxeter group is the symmetry group of the regular paving of the plane by squares and contains translations. Indeed, the product s_{2}s_{1}s_{2} is a reflection in the line parallel to the yaxis going through (1, 0) and thus the product t = s_{2} s_{1}s_{2}s_{3} is a translation by +2 in the xdirection. All powers of t are distinct; the group is infinite. This Coxeter group is of affine type and is called \(C_2^ +\) (which coincides with \(B_2^ +\))
3.2.2.2 The isomorphism problem
The Coxeter presentation of a given Coxeter group may not be unique. Consider for instance the group I_{2}(6) of order 12 of symmetries of the regular hexagon, generated by two reflections s_{1} and s_{2} with
This group is isomorphic with the rank 3 (reducible) Coxeter group I_{2} (3) × ℤ_{2}, with presentation
the isomorphism being given by f(r_{1}) = s_{1}, f(r_{2}) = s_{1}s_{2}s_{1}s_{2}s_{1}, f(r_{3}) = (s_{1}s_{2})^{3}. The question of determining all such isomorphisms between Coxeter groups is known as the “isomorphism problem of Coxeter groups”. This is a difficult problem whose general solution is not yet known [10].
3.2.3 The length function
An important concept in the theory of Coxeter groups is that of the length of an element. The length of \(w \in {\mathfrak C}\) is by definition the number of generators that appear in a minimal representation of w as a product of generators. Thus, if \(w = {s_{{i_1}}}\,{s_{{i_2}}}\, \cdots {s_{{i_l}}}\) and if there is no way to write w as a product of less than l generators, one says that w has length l.
For instance, for the dihedral group I_{2}(3), the identity has length zero, the generators s_{1} and s_{2} have length one, the two nontrivial rotations have length two, and the third reflection s_{3} has length three. Note that the rotations have representations involving two and four (and even a higher number of) generators since for instance s_{1}s_{2} = s_{2}s_{1}s_{2}s_{1}, but the length is associated with the representations involving as few generators as possible. There might be more than one such representation as it occurs for s_{3} = s_{1}s_{2}s_{1} = s_{2}s_{1} s_{2}. Both involve three generators and define the length of s_{3} to be three.
Let w be an element of length l. The length of ws_{i} (where s_{i} is one of the generators) differs from the length of w by an odd (positive or negative) integer since the relations among the generators always involve an even number of reflections. In fact, l(ws_{i}) is equal to l + 1 or l − 1 since l(ws_{i}) ≤ l(w) + 1 and l(w = ws_{i}s_{j}) ≤ l(ws_{i}) + 1. Thus, in ws_{i}, there can be at most one simplification (i.e., at most two elements that can be removed using the relations).
3.2.4 Geometric realization
We now construct a geometric realisation for any given Coxeter group. This enables one to view the Coxeter group as a group of linear transformations acting in a vector space of dimension n, equipped with a scalar product preserved by the group.
To each generator s_{i}, associate a vector α_{i} of a basis {α_{1}, …, α_{n}} of an ndimensional vector space V. Introduce a scalar product defined as follows,
on the basis vectors and extend it to V by linearity. Note that for i = j, m_{ii} = 1 implies B(α_{i},α_{i}) = 1 for all i. In the case of the dihedral group A_{2}, this scalar product is just the Euclidean scalar product in the twodimensional plane where the equilateral triangle lies, as can be seen by taking the two vectors α_{1} and α_{2} respectively orthogonal to the first and second lines of reflection in Figure 3 and oriented as indicated. But in general, the scalar product (3.28) might not be of Euclidean signature and might even be degenerate. This is the case for the infinite dihedral group I_{2}(∞), for which the matrix B reads
and has zero determinant. We shall occasionally use matrix notations for the scalar product, B(α,γ) = α^{T} Bγ.
However, the basis vectors are always all spacelike since they have norm squared equal to 1. For each i, the vector space V splits then as a direct sum
where H_{i} is the hyperplane orthogonal to α_{i} (δ ∈ H_{i} iff B(γ, α_{i}) =0). One defines the geometric reflection σ_{i} as
It is clear that σ_{i} fixes H_{i} pointwise and reverses α_{i}. It is also clear that \(\sigma _i^2 = 1\) and that σ_{i} preserves B,
Note that in the particular case of A_{2}, we recover in this way the reflections s_{1} and s_{2}.
We now verify that the σ_{i}’s also fulfill the relations \({({\sigma _i}{\sigma _j})^{{m_{ij}}}} = 1\). To that end we consider the plane Π spanned by α_{i} and α_{j}. This plane is left invariant under σ_{i} and σ_{j}. Two possibilities may occur:
 1.
The induced scalar product on Π is nondegenerate and in fact positive definite, or
 2.
the induced scalar product is positive semidefinite, i.e., there is a null direction orthogonal to any other direction.
The second case occurs only when m_{ij} = ∞. The null direction is given by γ = α_{i} + α_{j}.
In Case 1, V splits as \({\sigma _i}{\sigma _j}{)^{{m_{ij}}}}\) is clearly the identity on Π^{⊥} since both σ_{i} and σ_{j} leave Π^{⊥} pointwise invariant. One needs only to investigate \({\sigma _i}{\sigma _j}{)^{{m_{ij}}}}\) on Π, where the metric is positive definite. To that end we note that the reflections σ_{i} and σ_{j} are, on Π, standard Euclidean reflections in the lines orthogonal to α_{i} and α_{j}, respectively. These lines make an angle of π/m_{ij} and hence the product σ_{i}σ_{j} is a rotation by an angle of 2π/m_{ij}. It follows that \({({\sigma _i}{\sigma _j})^{{m_{ij}}}} = 1\) also on Π.
In Case 2, \({{m_{ij}}}\) is infinite and we must show that no power of the product σ_{i}σ_{j} gives the identity. This is done by exhibiting a vector γ for which (σ_{i}σ_{j})^{k}(γ) ≠ γ for all integers k different from zero. Take for instance α_{i}. Since one has (σ_{i}σ_{j})(α_{i}) = α_{i} + 2λ and (σ_{i}σ_{j})(λ) = λ, it follows that (σ_{i}σ_{j})^{k}(α_{i}) = α_{i} + 2kλ ≠ α_{i} unless k = 0.
As the defining relations are preserved, we can conclude that the map f from the Coxeter group generated by the s_{i}’s to the geometric group generated by the σ_{i}’s defined on the generators by f (s_{i}) = σ_{i} is a group homomorphism. We will show below that its kernel is the identity so that it is in fact an isomorphism.
Finally, we note that if the Coxeter graph is irreducible, as we assume, then the matrix B_{ij} is indecomposable. A matrix A_{ij} is called decomposable if after reordering of its indices, it decomposes as a nontrivial direct sum, i.e., if one can slit the indices i, j in two sets J and Λ such that A_{ij} = 0 whenever i ∈ J, j ∈ Λ or i ∈ Λ, j ∈ J. The indecomposability of B follows from the fact that if it were decomposable, the corresponding Coxeter graph would be disconnected as no line would join a point in the set Λ to a point in the set J.
3.2.5 Positive and negative roots
A root is any vector in the space V of the geometric realisation that can be obtained from one of the basis vectors σ_{i} by acting with an element w of the Coxeter group (more precisely, with its image f(w) under the above homomorphism, but we shall drop “f” for notational simplicity). Any root α can be expanded in terms of the α_{i}’s,
If the coefficients c_{i} are all nonnegative, we say that the root a is positive and we write α > 0. If the coefficients c_{i} are all nonpositive, we say that the root a is negative and we write α < 0. Note that we use strict inequalities here because if c_{i} = 0 for all i, then a is not a root. In particular, the α_{i}’s themselves are positive roots, called also “simple” roots. (Note that the simple roots considered here differ by normalization factors from the simple roots of KacMoody algebras, as we shall discuss below.) We claim that roots are either positive or negative (there is no root with some c_{i}’s in Equation (3.33) > 0 and some other c_{i}’s < 0). The claim follows from the fact that the image of a simple root by an arbitrary element w of the Coxeter group is necessarily either positive or negative.
This, in turn, is the result of the following theorem, which provides a useful criterion to tell whether the length l(ws_{i}) of ws_{i} is equal to l(w) + 1 or l(w) − 1.
Theorem: l(ws_{i}) = l(w) + 1 if and only if w(α_{i}) > 0.
The proof is given in [107], page 111.
It easily follows from this theorem that l(ws_{i}) = l(w) − 1 if and only if w(α_{i}) < 0. Indeed, l(ws_{i}) = l(w) − 1 is equivalent to l(w) = l(ws_{i}) + 1, i.e., l((ws_{i})s_{i}) = l(ws_{i}) + 1 and thus, by the theorem, ws_{i}(α_{i}) > 0. But since s_{i}(α_{i}) = −α_{i}, this is equivalent to w(α_{i}) < 0.
We have seen in Section 3.2.3 that there are only two possibilities for the length l(ws_{i}). It is either equal to l(w) + 1 or to l(w) − 1. From the theorem just seen, the root w(α_{i}) is positive in the first case and negative in the second. Since any root is the Coxeter image of one of the simple roots α_{i}, i.e., can be written as w(α_{i}) for some w and α_{i}, we can conclude that the roots are either positive or negative; there is no alternative.
The theorem can be used to provide a geometric interpretation of the length function. One can show [107] that l(w) is equal to the number of positive roots sent by w to negative roots. In particular, the fundamental reflection s associated with the simple root α_{s} maps α_{s} to its negative and permutes the remaining positive roots.
Note that the theorem implies also that the kernel of the homomorphism that appears in the geometric realisation of the Coxeter group is trivial. Indeed, assume f(w) = 1 where w is an element of the Coxeter group that is not the identity. It is clear that there exists one group generator s_{i} such that l(ws_{i}) = l(w) − 1. Take for instance the last generator occurring in a reduced expression of w. For this generator, one has w(α_{i}) < 0, which is in contradiction with the assumption f(w) = 1.
Because f is an isomorphism, we shall from now on identify the Coxeter group with its geometric realisation and make no distinction between s_{i} and σ_{i}.
3.2.6 Fundamental domain
In order to describe the action of the Coxeter group, it is useful to introduce the concept of fundamental domain. Consider first the case of the symmetry group A_{2} of the equilateral triangle. The shaded region \(\mathcal F\) in Figure 4 contains the vectors γ such that B(α_{1}, γ) ≥ 0 and B(α_{2}, γ) ≤ 0. It has the following important property: Any orbit of the group A_{2} intersects \({\mathcal F}\) once and only once. It is called for this reason a “fundamental domain”. We shall extend this concept to all Coxeter groups. However, when the scalar product B is not positive definite, there are inequivalent types of vectors and the concept of fundamental domain can be generalized a priori in different ways, depending on which region one wants to cover. (The entire space? Only the timelike vectors? Another region?) The useful generalization turns out not to lead to a fundamental domain of the action of the Coxeter group on the entire vector space V, but rather to a fundamental domain of the action of the Coxeter group on the socalled Tits cone \({\mathcal X}\), which is such that the inequalities B(α_{i}, γ) ≥ 0 continue to play the central role.
We assume that the scalar product is nondegenerate. Define for each simple root α_{i} the open halfspace
We define \({\mathcal E}\) to be the intersection of all A_{i},
This is a convex open cone, which is nonempty because the metric is nondegenerate. Indeed, as B is nondegenerate, one can, by a change of basis, assume for simplicity that the bounding hyperplanes B(α_{i}, γ) = 0 are the coordinate hyperplanes x_{i} = 0. \({\mathcal E}\) is then the region x_{i} > 0 (with appropriate orientation of the coordinates) and \({\mathcal F}\) is x_{i} ≥ 0. The closure
is then a closed convex cone^{Footnote 8}.
We next consider the union of the images of \(\mathcal F\) under the Coxeter group,
One can show [107] that this is also a convex cone, called the Tits cone. Furthermore, \({\mathcal F}\) is a fundamental domain for the action of the Coxeter group on the Tits cone; the orbit of any point in \({\mathcal X}\) intersects \({\mathcal F}\) once and only once [107]. The Tits cone does not coincide in general with the full space V and is discussed below in particular cases.
3.3 Finite Coxeter groups
An important class of Coxeter groups are the finite ones, like I_{2}(3) above. One can show that a Coxeter group is finite if and only if the scalar product defined by Equation (3.28) on V is Euclidean [107]. Finite Coxeter groups coincide with finite reflection groups in Euclidean space (through hyperplanes that all contain the origin) and are discrete subgroups of O(n). The classification of finite Coxeter groups is known and is given in Table 1 for completeness. For finite Coxeter groups, one has the important result that the Tits cone coincides with the entire space V [107].
3.4 Affine Coxeter groups
Affine Coxeter groups are by definition such that the bilinear form B is positive semidefinite but not positive definite. The radical V^{⊥} (defined as the subspace of vectors x for which B(x,y) = x^{T} By = 0 for all y) is then onedimensional (in the irreducible case). Indeed, since B is positive semidefinite, its radical coincides with the set N of vectors such that λ^{T}Bλ = 0 as can easily be seen by going to a basis in which B is diagonal (the eigenvalues of B are nonnegative). Furthermore, N is at least onedimensional since B is not positive definite (one of the eigenvalues is zero). Let μ be a vector in Vα = N. Let ν be the vector whose components are the absolute values of those of μ, ν_{i} = μ_{i}Because B_{ij} ≤ 0 for i ≠ j (see definition of B in Equation (3.28)), one has
and thus the vector ν belongs also to V^{⊥}. All the components of ν are strictly positive, ν_{i} > 0. Indeed, let J be the set of indices for which ν_{j} > 0 and I the set of indices for which ν_{i} = 0. From ∑_{j}B_{kj}ν_{j} = 0 (ν ∈ V^{⊥}) one gets, by taking k in I, that B_{ij} = 0 for all i ∈ I, j ∈ J, contrary to the assumption that the Coxeter system is irreducible (B is indecomposable). Hence, none of the components of any zero eigenvector μ can be zero. If V^{⊥} were more than onedimensional, one could easily construct a zero eigenvector of B with at least one component equal to zero. Hence, the eigenspace V^{⊥} of zero eigenvectors is onedimensional.
Affine Coxeter groups can be identified with the groups generated by affine reflections in Euclidean space (i.e., reflections through hyperplanes that may not contain the origin, so that the group contains translations) and have also been completely classified [107]. The translation subgroup of an affine Coxeter group \({\mathfrak C}\) is an invariant subgroup and the quotient \({{\mathfrak C}_0}\) is finite; the affine Coxeter group \({\mathfrak C}\) is equal to the semidirect product of its translation subgroup by \({{\mathfrak C}_0}\)_{0}. We list all the affine Coxeter groups in Table 2.
3.5 Lorentzian and hyperbolic Coxeter groups
Coxeter groups that are neither of finite nor of affine type are said to be of indefinite type. An important property of Coxeter groups of indefinite type is the following. There exists a positive vector (c_{i}) such that ∑_{j}B_{ij}c_{j} is negative [116]. A vector is said to be positive (respectively, negative) if all its components are strictly positive (respectively, strictly negative). This is denoted c_{i} > 0 (respectively, c_{i} < 0). Note that a vector may be neither positive nor negative, if some of its components are positive while some others are negative. Note also that these concepts refer to a specific basis. This property is demonstrated in Appendix A.
We assume, as already stated, that the scalar product B is nondegenerate. Let {ω_{i}} be the basis dual to the basis {α_{i}} in the scalar product B,
The ω_{i}’s are called “fundamental weights”. (The fundamental weights are really defined by Equation (3.38) up to normalization, as we will see in Section 3.6 on crystallographic Coxeter groups. They thus differ from the solutions of Equation (3.38) only by a positive multiplicative factor, irrelevant for the present discussion.)
Consider the vector υ = ∑_{i}c_{i}α_{i}, where the vector c_{i} is such that c_{i} > 0 and ∑_{j}B_{ij}c_{j} < 0. This vector exists since we assume the Coxeter group to be of indefinite type. Let Σ be the hyperplane orthogonal to υ. Because c_{i} > 0, the vectors ω_{i}’s all lie on the positive side of Σ, B(υ, ω_{i}) = c_{i} > 0. By contrast, the vectors α_{i}’s all lie on the negative side of Σ since B(α_{i}, υ) = ∑_{j}B_{ij}c_{j} < 0. Furthermore, υ has negative norm squared, B(υ, υ) =∑_{j}c_{j}(∑_{j}B_{ij}c_{j}) < 0. Thus, in the case of Coxeter groups of indefinite type (with a nondegenerate metric), one can choose a hyperplane such that the positive roots lie on one side of it and the fundamental weights on the other side. The converse is true for Coxeter group of finite type: In that case, there exists c_{i} > 0 such that ∑_{j}B_{ij}c_{j} is positive, implying that the positive roots and the fundamental weights are on the same side of the hyperplane Σ.
We now consider a particular subclass of Coxeter groups of indefinite type, called Lorentzian Coxeter groups. These are Coxeter groups such that the scalar product B is of Lorentzian signature (n − 1,1). They are discrete subgroups of the orthochronous Lorentz group O^{+}(n − 1, 1) preserving the time orientation. Since the α_{i} are spacelike, the reflection hyperplanes are timelike and thus the generating reflections s_{i} preserve the time orientation. The hyperplane Σ from the previous paragraph is spacelike. In this section, we shall adopt Lorentzian coordinates so that Σ has equation x^{0} = 0 and we shall choose the time orientation so that the positive roots have a negative time component. The fundamental weights have then a positive time component. This choice is purely conventional and is made here for convenience. Depending on the circumstances, the other time orientation might be more useful and will sometimes be adopted later (see for instance Section 4.8).
Turn now to the cone \({\mathcal E}\) defined by Equation (3.35). This cone is clearly given by
Similarly, its closure \({\mathcal F}\) is given by
The cone \({\mathcal F}\) is thus the convex hull of the vectors ω_{i}, which are on the boundary of \({\mathcal F}\).
By definition, a hyperbolic Coxeter group is a Lorentzian Coxeter group such that the vectors in \(\mathcal E\) are all timelike, B(λ, λ) < 0 for all \(\lambda \in {\mathcal E}\). Hyperbolic Coxeter groups are precisely the groups that emerge in the gravitational billiards of physical interest. The hyperbolicity condition forces B(λ, λ) < 0 for all \(\lambda \in {\mathcal F}\), and in particular, B(ω_{i}, ω_{i}) ≤ 0: The fundamental weights are timelike or null. The cone \({\mathcal F}\) then lies within the light cone. This does not occur for generic (nonhyperbolic) Lorentzian algebras.
The following theorem enables one to decide whether a Coxeter group is hyperbolic by mere inspection of its Coxeter graph.
Theorem: Let \({\mathfrak C}\) be a Coxeter group with irreducible Coxeter graph Γ. The Coxeter group is hyperbolic if and only if the following two conditions hold:
The bilinear form B is nondegenerate but not positive definite.
For each i, the Coxeter graph obtained by removing the node i from Γ is of finite or affine type.
(Note: By removing a node, one might get a nonirreducible diagram even if the original diagram is connected. A reducible diagram defines a Coxeter group of finite type if and only if each irreducible component is of finite type, and a Coxeter group of affine type if and only if each irreducible component is of finite or affine type with at least one component of affine type.)
Proof:
It is clear that if a Coxeter group is hyperbolic, then its bilinear form fulfills the first condition. Let ω_{i} be one of the vectors of the dual basis. The vectors α_{j} with j ≠ i form a basis of the hyperplane Π_{i} orthogonal to ω_{i}. Because ω_{i} is nonspacelike (the group is hyperbolic), the hyperplane Π_{i} is spacelike or null. The Coxeter graph defined by the α_{j} with j ≠ i (i.e., by removing the node α_{i}) is thus of finite or affine type.
Conversely, assume that the two conditions of the theorem hold. From the first condition, it follows that the set N = {λ ∈ V  B(λ, λ) < 0} is nonempty. Let Π_{i} be the hyperplane spanned by the α_{j} with j ≠ i, i.e., orthogonal to ω_{i}. From the second condition, it follows that the intersection of N with each Π_{i} is empty. Accordingly, each connected component of N lies in one of the connected components of the complement of \(\bigcup\nolimits_i {{\Pi _i}}\) Π_{i}, namely, is on a definite (positive or negative) side of each of the hyperplanes Π_{i}. These sets are of the form ∈_{i}c_{i}α_{i} with c_{i} > 0 for some i’s (fixed throughout the set) and c_{i} < 0 for the others. This forces the signature of B to be Lorentzian since otherwise there would be at least a twodimensional subspace Z of V such that Z {0} ⊂ N. Because Z {0} is connected, it must lie in one of the subsets just described. But this is impossible since if λ ∈ Z {0}, then −λ ∈ Z {0}.
We now show that \({\mathcal E} \subset N\). Because the signature of B is Lorentzian, N is the inside of the standard light cone and has two components, the “future” component and the “past” component. From the second condition of the theorem, each ω_{i} lies on or inside the light cone since the orthogonal hyperplane is nontimelike. Furthermore, all the ω_{i}’s are future pointing, which implies that the cone \({\mathcal E}\) lies in N, as had to be shown (a positive sum of future pointing non spacelike vectors is nonspacelike). This concludes the proof of the theorem.
In particular, this theorem is useful for determining all hyperbolic Coxeter groups once one knows the list of all finite and affine ones. To illustrate its power, consider the Coxeter diagram of Figure 8, with 8 nodes on the loop and one extra node attached to it (we shall see later that it is called \(A_7^{+ +}\)).
The bilinear form is given by
and is of Lorentzian signature. If one removes the node labelled 9, one gets the affine diagram \(A_7^{+}\) (see Figure 9). If one removes the node labelled 8, one gets the finite diagram of the direct product group A_{1} × A_{7} (see Figure 10). Deleting the nodes labelled 1 or 7 yields the finite diagram of A_{8}(see Figure 11). Removing the nodes labelled 2 or 6 gives the finite diagram of D_{8} (see Figure 12). If one removes the nodes labelled 3 or 5, one obtains the finite diagram of E_{8} (see Figure 13). Finally, deleting the node labelled 4 yields the affine diagram of \(E_7^{+}\) (see Figure 14). Hence, the Coxeter group is hyperbolic.
Consider now the same diagram, with one more node in the loop \((A_8^{+ +})\). In that case, if one removes one of the middle nodes 4 or 5, one gets the Coxeter group \(E_7^{+ +}\), which is neither finite nor affine. Hence, \(A_8^{+ +}\) is not hyperbolic.
Using the two conditions in the theorem, one can in fact provide the list of all irreducible hyperbolic Coxeter groups. The striking fact about this classification is that hyperbolic Coxeter groups exist only in ranks 3 ≤ n ≤ 10, and, moreover, for 4 ≤ n ≤ 10 there is only a finite number. In the n =3 case, on the other hand, there exists an infinite class of hyperbolic Coxeter groups. In Figure 15 we give a general form of the Coxeter graphs corresponding to all rank 3 hyperbolic Coxeter groups, and in Tables 3–9 we give the complete classification for 4 ≤ n ≤ 10.
Note that the inverse metric (B^{−1})_{ij}, which gives the scalar products of the fundamental weights, has only negative entries in the hyperbolic case since the scalar product of two futurepointing nonspacelike vectors is strictly negative (it is zero only when the vectors are both null and parallel, which does not occur here).
One can also show [116, 107] that in the hyperbolic case, the Tits cone \({\mathcal X}\) coincides with the future light cone. (In fact, it coincides with either the future light cone or the past light cone. We assume that the time orientation in V has been chosen as in the proof of the theorem, so that the Tits cone coincides with the future light cone.) This is at the origin of an interesting connection with discrete reflection groups in hyperbolic space (which justifies the terminology). One may realize hyperbolic space \({{\mathcal H}_{n  1}}\) as the upper sheet of the hyperboloid B(λ, λ) = −1 in V. Since the Coxeter group is a subgroup of O^{+}(n − 1,1), it leaves this sheet invariant and defines a group of reflections in \({{\mathcal H}_{n  1}}\). The fundamental reflections are reflections through the hyperplanes in hyperbolic space obtained by taking the intersection of the Minkowskian hyperplanes B(α_{i}, λ) = 0 with hyperbolic space. These hyperplanes bound the fundamental region, which is the domain to the positive side of each of these hyperplanes. The fundamental region is a simplex with vertices \({{\bar \omega}_i}\), where \({{\bar \omega}_i}\) are the intersection points of the lines ℝω_{i} with hyperbolic space. This intersection is at infinity in hyperbolic space if ω_{i} is lightlike. The fundamental region has finite volume but is compact only if the ω_{i} are timelike.
Thus, we see that the hyperbolic Coxeter groups are the reflection groups in hyperbolic space with a fundamental domain which (i) is a simplex, and which (ii) has finite volume. The fact that the fundamental domain is a simplex (n vectors in \({{\mathcal H}_{n  1}}\)) follows from our geometric construction where it is assumed that the n vectors α_{i} form a basis of V.
The group PGL(2, ℤ) relevant to pure gravity in four dimensions is easily verified to be hyperbolic.
For general information, we point out the following facts:
Compact hyperbolic Coxeter groups (i.e., hyperbolic Coxeter groups with a compact fundamental region) exist only for ranks 3, 4 and 5, i.e., in two, three and fourdimensional hyperbolic space. All hyperbolic Coxeter groups of rank > 5 have a fundamental region with at least one vertex at infinity. The hyperbolic Coxeter groups appearing in gravitational theories are always of the noncompact type.
There exist reflection groups in hyperbolic space whose fundamental domains are not simplices. Amazingly enough, these exist only in hyperbolic spaces of dimension < 995. If one imposes that the fundamental domain be compact, these exist only in hyperbolic spaces of dimension < 29. The bound can probably be improved [164].
Nonhyperbolic Lorentzian Coxeter groups are associated through the above construction with infinitevolume fundamental regions since some of the vectors ω_{i} are spacelike, which imply that the corresponding reflection hyperplanes intersect beyond hyperbolic infinity.
3.6 Crystallographic Coxeter groups
Among the Coxeter groups, only those that are crystallographic correspond to Weyl groups of KacMoody algebras. Therefore we now introduce this important concept. By definition, a Coxeter group is crystallographic if it stabilizes a lattice in V. This lattice need not be the lattice generated by the α_{i}’s. As discussed in [107], a Coxeter group is crystallographic if and only if two conditions are satisfied: (i) The integers m_{ij} (i ≠ j) are restricted to be in the set {2, 3, 4, 6, ∞}, and (ii) for any closed circuit in the Coxeter graph of \({\mathfrak C}\), the number of edges labelled 4 or 6 is even.
Given a crystallographic Coxeter group, it is easy to exhibit a lattice L stabilized by it. We can construct a basis for that lattice as follows. The basis vectors μ_{i} of the lattice are multiples of the original simple roots, μ_{i} = c_{i}α_{i} for some scalars c_{i} which we determine by applying the following rules:
\({m_{ij}} = 3 \Rightarrow {c_i} = {c_j}\).
\({m_{ij}} = 4 \Rightarrow {c_i} = \sqrt 2 {c_j}\,{\rm{or}}\,{c_j} = \sqrt 2 {c_j}\).
\({m_{ij}} = 6 \Rightarrow {c_i} = \sqrt 3 {c_j}\,{\rm{or}}\,{c_j} = \sqrt 3 {c_j}\).
\({m_{ij}} = \infty \Rightarrow {c_i} = {c_j}\).
One easily verifies that σ_{i}(μ_{j}) = μ_{j} − d_{ij}μ_{i} for some integers d_{ij}. Hence L is indeed stabilized. The integers d_{ij} are equal to \(2{{B({\mu _i},\,{\mu _j})} \over {B({\mu _i},\,{\mu _i})}}\).
The rules are consistent as can be seen by starting from an arbitrary node, say α_{1}, for which one takes c_{1} = 1. One then proceeds to the next nodes in the (connected) Coxeter graph by applying the above rules. If there is no closed circuit, there is no consistency problem since there is only one way to proceed from α_{1} to any given node. If there are closed circuits, one must make sure that one comes back to the same vector after one turn around any circuit. This can be arranged if the number of steps where one multiplies or divides by \(\sqrt 2\) (respectively, \(\sqrt 3\)) is even.
Our construction shows that the lattice L is not unique. If there are only two different lengths for the lattice vectors μ_{i} it is convenient to normalize the lengths so that the longest lattice vectors have length squared equal to two. This choice simplifies the factors \(2{{B({\mu _i},\,{\mu _j})} \over {B({\mu _i},\,{\mu _i})}}\).
The rank 10 hyperbolic Coxeter groups are all crystallographic. The lattices preserved by E_{10} and DE_{10} are unique up to an overall rescaling because the nontrivial m_{ij} (i ≠ j) are all equal to 3 and there is no choice in the ratios c_{i}/c_{j}, all equal to one (first rule above). The Coxeter group BE_{10} preserves two (dual) lattices.
3.6.1 On the normalization of roots and weights in the crystallographic case
Since the vectors μ_{i} and α_{i} are proportional, they generate identical reflections. Even though they do not necessarily have length squared equal to unity, the vectors μ_{i} are more convenient to work with because the lattice preserved by the Coxeter group is simply the lattice ∈_{i}ℤμ_{i} of points with integer coordinates in the basis {μ_{i}}. For this reason, we shall call from now on “simple roots” the vectors μ_{i} and, to follow common practice, will sometimes even rename them α_{i}. Thus, in the crystallographic case, the (redefined) simple roots are appropriately normalized to the lattice structure. It turns out that it is with this normalization that simple roots of Coxeter groups correspond to simple roots of KacMoody algebras defined in the Section 4.6.3. A root is any point on the root lattice that is in the Coxeter orbit of some (redefined) simple root. It is these roots that coincide with the (real) roots of KacMoody algebras.
It is also useful to rescale the fundamental weights. The rescaled fundamental weights, of course proportional to ω_{i}, are denoted Λ_{i}. The convenient normalization is such that
With this normalization, they coincide with the fundamental weights of KacMoody algebras, to be considered in Section 4.
4 Lorentzian Kac—Moody Algebras
The explicit appearance of infinite crystallographic Coxeter groups in the billiard limit suggests that gravitational theories might be invariant under a huge symmetry described by Lorentzian KacMoody algebras (defined in Section 4.1). Indeed, there is an intimate connection between crystallographic Coxeter groups and KacMoody algebras. This connection might be familiar in the finite case. For instance, it is well known that the finite symmetry group A_{2} of the equilateral triangle (isomorphic to the group of permutations of 3 objects) and the corresponding hexagonal pattern of roots are related to the finitedimensional Lie algebra \(\mathfrak {sl}(3,\,\mathbb R)\) (or \(\mathfrak {su}\)(3)). The group A_{2} is in fact the Weyl group of \(\mathfrak {sl}(3,\,\mathbb R)\) (see Section 4.7).
This connection is not peculiar to the Coxeter group A2 but is generally valid: Any crystallographic Coxeter group is the Weyl group of a KacMoody algebra traditionally denoted in the same way (see Section 4.7). This is the reason why it is expected that the Coxeter groups might signal a bigger symmetry structure. And indeed, there are indications that this is so since, as we shall discuss in Section 9, an attempt to reformulate the gravitational Lagrangians in a way that makes the conjectured symmetry manifest yields intriguing results.
The purpose of this section is to develop the mathematical concepts underlying KacMoody algebras and to explain the connection between Coxeter groups and KacMoody algebras. How this is relevant to gravitational theories will be discussed in Section 5.
4.1 Definitions
An n × n matrix A is called a “generalized Cartan matrix” (or just “Cartan matrix” for short) if it satisfies the following conditions^{Footnote 9}:
where ℤ_ denotes the nonpositive integers. One can encode the Cartan matrix in terms of a Dynkin diagram, which is obtained as follows:
 1.
For each i = 1, …, n, one associates a node in the diagram.
 2.
One draws a line between the node i and the node j if A_{ij} ≠ 0; if A_{ij} = 0 (= A_{ij}), one draws no line between i and j.
 3.
One writes the pair (A_{ij}, A_{ij}) over the line joining i to j. When the products A_{ij} · A_{ij} are all ≤ 4 (which is the only situation we shall meet in practice), this third rule can be replaced by the following rules:
 (a)
one draws a number of lines between i and j equal to max( A_{ij},  A_{ij});
 (b)
one draws an arrow from j to i if A_{ij}  > A_{ij}.
 (a)
So, for instance, the Dynkin diagrams in Figure 16 correspond to the Cartan matrices
respectively. If the Dynkin diagram is connected, the matrix A is indecomposable. This is what shall be assumed in the following.
Although this is not necessary for developing the general theory, we shall impose two restrictions on the Cartan matrix. The first one is that det A ≠ 0; the second one is that A is symmetrizable. The restriction det A ≠ 0 excludes the important class of affine algebras and will be lifted below. We impose it at first because the technical definition of the KacMoody algebra when det A = 0 is then slightly more involved.
The second restriction imposes that there exists an invertible diagonal matrix D with positive elements ϵ_{i} and a symmetric matrix S such that
The matrix S is called a symmetrization of A and is unique up to an overall positive factor because A is indecomposable. To prove this, choose the first (diagonal) element ϵ_{1} > 0 of D arbitrarily. Since A is indecomposable, there exists a nonempty set J_{1} of indices j such that A_{1j} ≠ 0. One has A_{1j} = ϵ_{1}S_{1j} and A_{j1} = ϵ_{j}S_{j1}. This fixes the ϵ_{j}’s > 0 in terms of ϵ_{1} since S_{1j} = S_{j1}. If not all the elements ϵ_{j} are determined at this first step, we pursue the same construction with the elements A_{jk} = ϵ_{j}S_{jk} and A_{kj} = ϵ_{k}S_{kj} = ϵ_{k}S_{kj} with j ϵ J_{1} and, more generally, at step p, with j ∈ J_{1} ∩ J_{2} ⋯ ∩ J_{p}. As the matrix A is assumed to be indecomposable, all the elements ϵ_{i} of D and S_{ij} of S can be obtained, depending only on the choice of ϵ_{1}. One gets no contradicting values for the ϵ_{j}’s because the matrix A is assumed to be symmetrizable.
In the symmetrizable case, one can characterize the Cartan matrix according to the signature of (any of) its symmetrization(s). One says that A is of finite type if S is of Euclidean signature, and that it is of Lorentzian type if S is of Lorentzian signature.
Given a Cartan matrix A (with det A ≠ 0), one defines the corresponding KacMoody algebra \(\mathfrak {g}=\mathfrak {g}(A)\) as the algebra generated by 3n generators h_{i}, e_{i}, f_{i} subject to the following “ChevalleySerre” relations (in addition to the Jacobi identity and antisymmetry of the Lie bracket),
The relations (4.11), called Serre relations, read explicitly
(and likewise for the f_{k}’s).
Any multicommutator can be reduced, using the Jacobi identity and the above relations, to a multicommutator involving only the e_{i}’s, or only the f_{i}’s. Hence, the KacMoody algebra splits as a direct sum (“triangular decomposition”)
where \({\mathfrak {n}_ }\) is the subalgebra involving the multicommutators \([{f_{{i_1}}},\,[{f_{{i_2}}},\, \cdots, \,[{f_{{i_{k  1}}}},\,{f_{{i_k}}}] \cdots ],\,{\mathfrak {n}_ +}\) is the subalgebra involving the multicommutators \([{e_{{i_1}}},\,[{e_{{i_2}}},\, \cdots, \,[{e_{{i_{k  1}}}},\,{e_{{i_k}}}] \cdots ]\) and \({\mathfrak h}\) is the Abelian subalgebra containing the h_{i}’s. This is called the Cartan subalgebra and its dimension n is the rank of the KacMoody algebra \({\mathfrak g}\). It should be stressed that the direct sum Equation (4.13) is a direct sum of \({\mathfrak {n}_ }\), \({\mathfrak h}\) and \({\mathfrak {n}_ +}\) as vector spaces, not as subalgebras (since these subalgebras do not commute).
A priori, the numbers of the multicommutators
are infinite, even after one has taken into account the Jacobi identity. However, the Serre relations impose nontrivial relations among them, which, in some cases, make the KacMoody algebra finitedimensional. Three worked examples are given in Section 4.4 to illustrate the use of the Serre relations. In fact, one can show [116] that the KacMoody algebra is finitedimensional if and only if the symmetrization S of A is positive definite. In that case, the algebra is one of the finitedimensional simple Lie algebras given by the Cartan classification. The list is given in Table 10.
When the Cartan matrix A is of Lorentzian signature the KacMoody algebra \({\mathfrak {g}(A)}\), constructed from A using the ChevalleySerre relations, is called a Lorentzian KacMoody algebra. This is the case of main interest for the remainder of this paper.
4.2 Roots
The adjoint action of the Cartan subalgebra on \({\mathfrak {n}_ +}\) and \({\mathfrak {n}_ }\) is diagonal. Explicitly,
for any element \(h \in {\mathfrak h}\), where α_{i} is the linear form on \({\mathfrak h}\) (i.e., the element of the dual \({\mathfrak h^{\ast}}\)) defined by α_{i}(h_{j}) = A_{ji}. The α_{i}’s are called the simple roots. Similarly,
and, if \([{e_{{i_1}}},\,[{e_{{i_2}}},\, \cdots, \,[{e_{{i_{k  1}}}},\,{e_{{i_k}}}] \cdots ]\) is nonzero, one says that \({\alpha _{{i_1}}} + {\alpha _{{i_2}}} + \cdots {\alpha _{{i_k}}}\) is a positive root. On the negative side, \({\mathfrak {n}_ }\), one has
and \( ({\alpha _{{i_1}}} + {\alpha _{{i_2}}} + \cdots {\alpha _{{i_k}}})(h)\) is called a negative root when \([{f_{{i_1}}},\,[{f_{{i_2}}},\, \cdots, \,[{f_{{i_{k  1}}}},\,{f_{{i_k}}}]\) is nonzero. This occurs if and only if \([{e_{{i_1}}},[{e_{{i_2}}}, \cdots ,[{e_{{i_{k  1}}}},{e_{{i_k}}}] \cdots ]\) is nonzero: −α is a negative root if and only if α is a positive root.
We see from the construction that the roots (linear forms α such that [h, x] = α(h)x has nonzero solutions x) are either positive (linear combinations of the simple roots α_{i} with integer nonnegative coefficients) or negative (linear combinations of the simple roots with integer nonpositive coefficients). The set of positive roots is denoted by Δ_{+}; that of negative roots by Δ_{−}. The set of all roots is Δ, so we have Δ = Δ_{+} ∪Δ_{−}. The simple roots are positive and form a basis of \({\mathfrak h^{\ast}}\). One sometimes denotes the h_{i} by \(\alpha _i^ \vee\) (and thus, \([\alpha _i^ \vee, \,{e_j}] = {A_{ij}}{e_j}\) etc). Similarly, one also uses the notation 〈·,·〉 for the standard pairing between \({\mathfrak h}\) and its dual \({\mathfrak h^{\ast}}\), i.e., 〈α, h〉 = α(h). In this notation the entries of the Cartan matrix can be written as
Finally, the root lattice Q is the set of linear combinations with integer coefficients of the simple roots,
All roots belong to the root lattice, of course, but the converse is not true: There are elements of Q that are not roots.
4.3 The Chevalley involution
The symmetry between the positive and negative subalgebras \({\mathfrak {n}_ +}\) and \({\mathfrak {n}_ }\) of the KacMoody algebra can be rephrased formally as follows: The KacMoody algebra is invariant under the Chevalley involution τ, defined on the generators as
The Chevalley involution is in fact an algebra automorphism that exchanges the positive and negative sides of the algebra.
Finally, we quote the following useful theorem.
Theorem: The KacMoody algebra \({\mathfrak g}\) defined by the relations (4.10, 4.11) is simple. The proof may be found in Kac’ book [116], page 12.
We note that invertibility and indecomposability of the Cartan matrix A are central ingredients in the proof. In particular, the theorem does not hold in the affine case, for which the Cartan matrix is degenerate and has nontrivial ideals^{Footnote 10} (see [116] and Section 4.5).
4.4 Three examples
To get a feeling for how the Serre relations work, we treat in detail three examples.
A_{2}: We start with A_{2}, the Cartan matrix of which is Equation (4.4). The defining relations are then:
$$\begin{array}{*{20}c} {[{h_1},{h_2}] = 0,\quad \;\;} & {[{h_1},{e_1}] = 2{e_1},\;\;} & {[{h_1},{e_2}] =  {e_2},\quad} \\ {[{h_1},{f_1}] =  2{f_1},} & {[{h_1},{f_2}] = {f_2},\quad} & {[{h_2},{e_1}] =  {e_1},\quad} \\ {[{h_2},{e_2}] = 2{e_2},\;\;} & {[{h_2},{f_1}] = {f_1},\quad} & {[{h_2},{f_2}] =  2{f_2},\;\;} \\ {[{e_1},[{e_1},{e_2}]] = 0,\;\;\quad \quad \;\;} & {[{e_2},[{e_2},{e_1}]] = 0,\quad \quad \quad} & {[{f_1},[{f_1},{f_2}]] = 0,\quad \quad \quad \quad} \\ {[{f_2},[{f_2},{f_1}]] = 0\;\;\quad \quad \quad \;} & {[{e_i},{f_j}] = {\delta _{ij}}{h_j}.} & {} \\ \end{array}$$(4.20)The commutator [e_{1}, e_{2}] is not killed by the defining relations and hence is not equal to zero (the defining relations are all the relations). All the commutators with three (or more) e’s are however zero. A similar phenomenon occurs on the negative side. Hence, the algebra A_{2} is eightdimensional and one may take as basis {h_{1}, h_{2}, e_{1}, e_{2}, [e_{1}, e_{2}], f_{1}, f_{2}, [f_{1}, f_{2}]}. The vector [e_{1}, e_{2}] corresponds to the positive root α_{1} + α_{2}.
B_{2}: The algebra B_{2}, the Cartan matrix of which is Equation (4.5), is defined by the same set of generators, but the Serre relations are now [e_{1}, [e_{1}, [e_{1}, e_{2}]]] = 0 and [e_{2}, [e_{2}, e_{1}]] = 0 (and similar relations for the f’s). The algebra is still finitedimensional and contains, besides the generators, the commutators [e_{1}, e_{2}], [e_{1}, [e_{1}, e_{2}]], their negative counterparts [f_{1}, f_{2}] and [f_{1}, [f_{1}, f_{2}]], and nothing else. The triple commutator [e_{1}, [e_{1}, [e_{1}, e_{2}]]] vanishes by the Serre relations. The other triple commutator [e_{2}, [e_{1}, [e_{1}, e_{2}]]] vanishes also by the Jacobi identity and the Serre relations,
$$[{e_1},[{e_1},[{e_1},{e_2}]]] = [[{e_2},{e_1}],[{e_1},{e_2}]] + [{e_1},[{e_2},[{e_1},{e_2}]]] = 0.$$(Each term on the righthand side is zero: The first by antisymmetry of the bracket and the second because [e_{2}, [e_{1}, e_{2}]] = −[e_{2}, [e_{2}, e_{1}]] = 0.) The algebra is 10dimensional and is isomorphic to \({\mathfrak {so}(3,\,2)}\).
\(A_1^{+}\): We now turn to \(A_1^{+}\), the Cartan matrix of which is Equation (4.8). This algebra is defined by the same set of generators as A_{2}, but with Serre relations given by
$$\begin{array}{*{20}c} {[{e_1},[{e_1},[{e_1},{e_2}]]] = 0,}\\ {[{e_2},[{e_2},[{e_2},{e_1}]]] = 0}\\\end{array}$$(4.21)(and similar relations for the f’s). This innocentlooking change in the Serre relations has dramatic consequences because the corresponding algebra is infinitedimensional. (We analyze here the algebra generated by the h’s, e’s and f’s, which is in fact the derived KacMoody algebra — see Section 4.5 on affine KacMoody algebras. The derived algebra is already infinitedimensional.) To see this, consider the \({\mathfrak {sl}(2,\,\mathbb R)}\) current algebra, defined by
$$[J_m^a,J_n^b] = {f^{ab}}_cJ_{m + n}^c + m{k^{ab}}c{\delta _{m + n,0}},$$(4.22)where \(a = 3,\, +, \, , \,{f^{ab}}_c\) are the structure constants of \({\mathfrak {sl}(2,\,\mathbb R)}\) and where k^{ab} is the invariant metric on \({\mathfrak {sl}(2,\,\mathbb R)}\) which we normalize here so that k^{−+} = 1. The subalgebra with n = 0 is isomorphic to \({\mathfrak {sl}(2,\,\mathbb R)}\),
$$[J_0^3,J_0^ + ] = 2J_0^ + ,\qquad [J_0^3,J_0^  ] =  2J_0^  ,\qquad [J_0^ + ,J_0^  ] = J_0^3.$$The current algebra (4.22) is generated by \(J_0^a\), c, \(J_0^ \) and \(J_{ 1}^ +\) since any element can be written as a multicommutator involving them. The map
$$\begin{array}{*{20}c} {{h_1}} & \rightarrow & {J_0^3,} & {{h_2}} & \rightarrow & { J_0^3 + c,} \\ {{e_1}} & \rightarrow & {J_0^ + ,} & {{e_2}} & \rightarrow & {J_1^  ,} \\ {{f_1}} & \rightarrow & {J_0^  ,} & {{f_2}} & \rightarrow & {J_{ 1}^ +} \\ \end{array}$$(4.23)preserves the defining relations of the KacMoody algebra and defines an isomorphism of the (derived) KacMoody algebra with the current algebra. The KacMoody algebra is therefore infinitedimensional. One can construct nonvanishing infinite multicommutators, in which e_{1} and e_{2} alternate:
$$\begin{array}{*{20}c} {\left[ {{e_1},} \right.\left[ {{e_2},} \right.\left[ {{e_1}, \cdots ,\left[ {{e_1},{e_2}} \right]\left. {\left. \cdots \right]} \right]} \right]\sim J_n^3\;\;\;} & {(n\;{e_1}\prime{\rm{s}}\;{\rm{and}}\;n\;{e_2}\prime{\rm{s}}),\;\;\;\;} \\ {\left[ {{e_1},} \right.\left[ {{e_2},} \right.\left[ {{e_1}, \cdots ,\left[ {{e_2},{e_1}} \right]\left. {\left. \cdots \right]} \right]} \right]\sim J_n^ + \;\;} & {(n + 1\;{e_1}\prime{\rm{s}}\;{\rm{and}}\;n\;{e_2}\prime{\rm{s}}),\;} \\ {\left[ {{e_2},} \right.\left[ {{e_1},} \right.\left[ {{e_2}, \cdots ,\left[ {{e_1},{e_2}} \right]\left. {\left. \cdots \right]} \right]} \right]\sim J_{n + 1}^ } & {(n\;{e_1}\prime{\rm{s}}\;{\rm{and}}\;n + 1\;{e_2}\prime{\rm{s}}).} \\ \end{array}$$(4.24)The Serre relations do not cut the chains of multicommutators to a finite number.
We see from these examples that the exact consequences of the Serre relations might be intricate to derive explicitly. This is one of the difficulties of the theory.
4.5 The affine case
The affine case is characterized by the conditions that the Cartan matrix has vanishing determinant, is symmetrizable and is such that its symmetrization S is positive semidefinite (only one zero eigenvalue). As before, we also take the Cartan matrix to be indecomposable. By a reasoning analogous to what we did in Section 3.4 above, one can show that the radical of S is onedimensional and that the ranks of S and A are equal to n − 1.
One defines the corresponding KacMoody algebras in terms of 3n + 1 generators, which are the same generators h_{i}, e_{i}, f_{i} subject to the same conditions (4.10, 4.11) as above, plus one extra generator η which can be taken to fulfill
The algebra admits the same triangular decomposition as above,
but now the Cartan subalgebra \({\mathfrak h}\) has dimension n + 1 (it contains the extra generator n).
Because the matrix A_{ij} has vanishing determinant, one can find a_{i} such that ∑_{i}a_{i}A_{ij} = 0. The element c = ∑_{i}a_{i}h_{i} is in the center of the algebra. In fact, the center of the KacMoody algebra is onedimensional and coincides with ℂc [116]. The derived algebra \({\mathfrak {g}{\prime}} = [\mathfrak {g},\,\mathfrak {g}]\) is the subalgebra generated by h_{i}, e_{i}, f_{i} and has codimension one (it does not contain η). One has
(direct sum of vector spaces, not as algebras). The only proper ideals of the affine KacMoody algebra \({\mathfrak g}\) are \({\mathfrak {g}{\prime}}\) and ℂc.
Affine KacMoody algebras appear in the BKL context as subalgebras of the relevant Lorentzian KacMoody algebras. Their complete list is known and is given in Tables 11 and 12.
4.6 The invariant bilinear form
4.6.1 Definition
To proceed, we assume, as announced above, that the Cartan matrix is invertible and symmetrizable since these are the only cases encountered in the billiards. Under these assumptions, an invertible, invariant bilinear form is easily defined on the algebra. We denote by ϵ_{i} the diagonal elements of D,
First, one defines an invertible bilinear form in the dual \({\mathfrak h^{\ast}}\) of the Cartan subalgebra. This is done by simply setting
for the simple roots. It follows from A_{ii} = 2 that
and thus the Cartan matrix can be written as
It is customary to fix the normalization of S so that the longest roots have (α_{i}α_{i}) = 2. As we shall now see, the definition (4.29) leads to an invariant bilinear form on the KacMoody algebra.
Since the bilinear form (··) is nondegenerate on \({\mathfrak h^{\ast}}\), one has an isomorphism \(\mu : \mathfrak {h}^{\ast} \rightarrow \mathfrak {h}\) defined by
This isomorphism induces a bilinear form on the Cartan subalgebra, also denoted by (··). The inverse isomorphism is denoted by v and is such that
Since the Cartan elements \({h_i} \equiv \alpha _i^ \vee\) obey
it is clear from the definitions that
and thus also
The bilinear form (··) can be uniquely extended from the Cartan subalgebra to the entire algebra by requiring that it is invariant, i.e., that it fulfills
for instance, for the e_{i}’s and f_{i}’s one finds
and similarly
In the same way we have
and thus
Quite generally, if e_{α} and e_{γ} are root vectors corresponding respectively to the roots α and γ,
then (e_{α}e_{γ}) = 0 unless γ = −α. Indeed, one has
and thus
It is proven in [116] that the invariance condition on the bilinear form defines it indeed consistently and that it is nondegenerate. Furthermore, one finds the relations
4.6.2 Real and imaginary roots
Consider the restriction (··)_{ℝ} of the bilinear form to the real vector space \(\mathfrak {h} _{\mathbb {R}}^ {\ast}\) obtained by taking the real span of the simple roots,
This defines a scalar product with a definite signature. As we have mentioned, the signature is Euclidean if and only if the algebra is finitedimensional [116]. In that case, all roots — and not just the simple ones — are spacelike, i.e., such that (αα) > 0.
When the algebra is infinitedimensional, the invariant scalar product does not have Euclidean signature. The spacelike roots are called “real roots”, the nonspacelike ones are called “imaginary roots” [116]. While the real roots are nondegenerate (i.e., the corresponding eigenspaces, called “root spaces”, are onedimensional), this is not so for imaginary roots. In fact, it is a challenge to understand the degeneracy of imaginary roots for general indefinite KacMoody algebras, and, in particular, for Lorentzian KacMoody algebras.
Another characteristic feature of real roots, familiar from standard finitedimensional Lie algebra theory, is that if α is a (real) root, no multiple of α is a root except ±α. This is not so for imaginary roots, where 2α (or other nontrivial multiples of α) can be a root even if α is. We shall provide explicit examples below.
Finally, while there are at most two different root lengths in the finitedimensional case, this is no longer true even for real roots in the case of infinitedimensional KacMoody algebras^{Footnote 11}. When all the real roots have the same length, one says that the algebra is “simplylaced”. Note that the imaginary roots (if any) do not have the same length, except in the affine case where they all have length squared equal to zero.
4.6.3 Fundamental weights and the Weyl vector
The fundamental weights {Λ_{i}} of the KacMoody algebra are vectors in the dual space \({\mathfrak h^{\ast}}\) of the Cartan subalgebra defined by
This implies
The Weyl vector ρ ∈ \(\rho \in {\mathfrak h^{\ast}}\) is defined by
and is thus equal to
4.6.4 The generalized Casimir operator
From the invariant bilinear form, one can construct a generalized Casimir operator as follows.
We denote the eigenspace associated with α by \({\mathfrak {g}_\alpha}\). This is called the “root space” of α and is defined as
A representation of the KacMoody algebra is called restricted if for every vector v of the representation subspace V, one has \({\mathfrak {g}_\alpha \cdot \upsilon = 0}\) for all but a finite number of positive roots α.
Let \(\{e_\alpha ^K\}\) be a basis of \({\mathfrak {g}_\alpha}\) and let \(\{e_{ \alpha}^K\}\) be the basis of \({\mathfrak {g}_ { {\alpha}}}\) dual to \(\{e_\alpha ^K\}\) in the Bmetric,
Similarly, let {u_{i}} be a basis of \({\mathfrak {g}_\alpha}\) and {u^{i}} the dual basis of \({\mathfrak h}\) with respect to the bilinear form
We set
where ρ is the Weyl vector. Recall from Section 4.6.1 that μ is an isomorphism from \({\mathfrak h^{\star}}\) to \({\mathfrak h}\), so, since ρ ∈ \(\rho \in {\mathfrak h^{\star}}\), the expression μ(ρ) belongs to \({\mathfrak h}\) as required. When acting on any vector of a restricted representation, Ω is welldefined since only a finite number of terms are different from zero.
It is proven in [116] that Ω commutes with all the operators of any restricted representation. For that reason, it is known as the (generalized) Casimir operator. It is quadratic in the generators^{Footnote 12}.
4.6.4.1 Note
This definition — and, in particular, the presence of the linear term μ(ρ) — might seem a bit strange at first sight. To appreciate it, turn to a finitedimensional simple Lie algebra. In the above notations, the usual expression for the quadratic Casimir operator reads
(without degeneracy index K since the roots are nondegenerate in the finitedimensional case). Here, κ^{AB} is the Killing metric and {T_{A}} a basis of the Lie algebra. This expression is not “normalordered” because there are, in the last term, lowering operators standing on the right. We thus replace the last term by
Using the fact that in a finitedimensional Lie algebra, \(\rho = \left({1/2} \right)\sum\nolimits_{\alpha \in {\Delta _ +}} \alpha\), (see, e.g., [85]) one sees that the Casimir operator can be rewritten in “normal ordered” form as in Equation (4.52). The advantage of the normalordered form is that it makes sense also for infinitedimensional KacMoody algebras in the case of restricted representations.
4.7 The Weyl group
The Weyl group \(\mathfrak{W}[\mathfrak{g}]\) of a KacMoody algebra \(\mathfrak{g}\) is a discrete group of transformations acting on \(\mathfrak{h}^{\ast}\). It is defined as follows. One associates a “fundamental Weyl reflection” \({r_i} \in \mathfrak{W}[\mathfrak{g}]\) to each simple root through the formula
The Weyl group is just the group generated by the fundamental Weyl reflections. In particular,
The Weyl group enjoys a number of interesting properties [116]:
It preserves the scalar product on \(\mathfrak{h}^{\ast}\).
It preserves the root lattice and hence is crystallographic.
Two roots that are in the same orbit have identical multiplicities.
Any real root has in its orbit (at least) one simple root and hence, is nondegenerate.
The Weyl group is a Coxeter group. The connection between the Coxeter exponents and the Cartan integers A_{ij} is given in Table 13 (i ≠ j).
This close relationship between Coxeter groups and KacMoody algebras is the reason for denoting both with the same notation (for instance, A_{n} denotes at the same time the Coxeter group with Coxeter graph of type A_{n} and the KacMoody algebra with Dynkin diagram A_{n}).
Note that different KacMoody algebras may have the same Weyl group. This is in fact already true for finitedimensional Lie algebras, where dual algebras (obtained by reversing the arrows in the Dynkin diagram) have the same Weyl group. This property can be seen from the fact that the Coxeter exponents are related to the dualityinvariant product A_{ij}A_{ji}. But, on top of this, one sees that whenever the product A_{ij}A_{ji} exceeds four, which occurs only in the infinitedimensional case, the Coxeter exponent m_{ij} is equal to infinity, independently of the exact value of A_{ij}A_{ji}. Information is thus clearly lost. For example, the Cartan matrices
lead to the same Weyl group, even though the corresponding KacMoody algebras are not isomorphic or even dual to each other.
Because the Weyl groups are (crystallographic) Coxeter groups, we can use the theory of Coxeter groups to analyze them. In the KacMoody context, the fundamental region is called “the fundamental Weyl chamber”.
We also note that by (standard vector space) duality, one can define the action of the Weyl group in the Cartan subalgebra \(\mathfrak{h}\), such that
One has using Equations (4.30, 4.32, 4.33, 4.35),
Finally, we leave it to the reader to verify that when the products A_{ij}A_{ji} are all ≤ 4, then the geometric action of the Coxeter group considered in Section 3.2.4 and the geometric action of the Weyl group considered here coincide. The (real) roots and the fundamental weights differ only in the normalization and, once this is taken into account, the metrics coincide. This is not the case when some products A_{ij}A_{ji} exceed 4. It should be also pointed out that the imaginary roots of the KacMoody algebras do not have immediate analogs on the Coxeter side.
4.7.1 Examples

Consider the Cartan matrices
$$A\prime = \left({\begin{array}{*{20}c} 2 & { 2} & 0 \\ { 2} & 2 & { 1} \\ 0 & { 1} & 2 \\ \end{array}} \right),\qquad A\prime \prime = \left({\begin{array}{*{20}c} 2 & { 4} & 0 \\ { 1} & 2 & { 1} \\ 0 & { 1} & 2 \\ \end{array}} \right)$$As the first (respectively, second) Cartan matrix defines the Lie algebra \(A_1^{+ +}\) (respectively \(A_2^{(2) +}\)) introduced below in Section 4.9, we also write it as \(A\prime \equiv A[A_1^{+ +}]\) (respectively, \(A\prime \prime \equiv A[A_2^{(2) +}]\)). We denote the associated sets of simple roots by {α′_{1}, α′_{2}, α′_{3}} and {α″_{1}, α″_{2}, α″_{3}}, respectively. In both cases, the Coxeter exponents are m_{12} = ∞, m_{13} = 2, m_{23} = 3 and the metric B_{ij} of the geometric Coxeter construction is
$$A\prime = \left({\begin{array}{*{20}c} 1 & { 1} & 0 \\ { 1} & 1 & { {1 \over 2}} \\ 0 & { {1 \over 2}} & 1 \\ \end{array}} \right).$$We associate the simple roots {α_{1}, α_{2}, α_{3}} with the geometric realisation of the Coxeter group \(\mathfrak{B}\) defined by the matrix B. These roots may a priori differ by normalizations from the simple roots of the KacMoody algebras described by the Cartan matrices A′ and A″.
Choosing the longest KacMoody roots to have squared length equal to two yields the scalar products
$$S\prime = \left({\begin{array}{*{20}c} 2 & { 2} & 0 \\ { 2} & 2 & { 1} \\ 0 & { 1} & 2 \\ \end{array}} \right),\qquad S\prime \prime = \left({\begin{array}{*{20}c} {{1 \over 2}} & { 1} & 0 \\ { 1} & 2 & { 1} \\ 0 & { 1} & 2 \\ \end{array}} \right).$$Recall now from Section 3 that the fundamental reflections \({\sigma _i} \in \mathfrak{B}\) have the following geometric realisation
$${\sigma _i}({\alpha _j}) = {\alpha _j}  2{B_{ij}}{\alpha _i}\qquad (i = 1,2,3),$$(4.60)which in this case becomes
$$\begin{array}{*{20}c} {{\sigma _1}:} & {{\alpha _1} \rightarrow  {\alpha _1},\quad \quad} & {{\alpha _2} \rightarrow {\alpha _2} + 2{\alpha _1},\;} & {{\alpha _3} \rightarrow {\alpha _3},\quad \quad} \\ {{\sigma _2}:} & {{\alpha _1} \rightarrow {\alpha _1} + 2{\alpha _2},} & {{\alpha _2} \rightarrow  {\alpha _2},\quad \quad} & {{\alpha _3} \rightarrow {\alpha _3} + {\alpha _2},} \\ {{\sigma _3}:} & {{\alpha _1} \rightarrow {\alpha _1},\quad \quad} & {{\alpha _2} \rightarrow {\alpha _2} + {\alpha _3},\;} & {{\alpha _3} \rightarrow  {\alpha _3}.\quad \;} \\\end{array}$$We now want to compare this geometric realisation of \(\mathfrak{B}\) with the action of the Weyl groups of A′ and A″ on the corresponding simple roots α′_{i} and α″_{i}. According to Equation (4.56), the Weyl group \(\mathfrak{W}[A_1^{+ +}]\) acts as follows on the roots α′_{i}
$$\begin{array}{*{20}c} {{{r\prime}_1}:} & {{{\alpha \prime}_1} \rightarrow  {{\alpha \prime}_1},\quad \quad} & {{{\alpha \prime}_2} \rightarrow {{\alpha \prime}_2} + 2{{\alpha \prime}_1},\;} & {{{\alpha \prime}_3} \rightarrow {{\alpha \prime}_3},\quad \quad} \\ {{{r\prime}_2}:} & {{{\alpha \prime}_1} \rightarrow {{\alpha \prime}_1} + 2{{\alpha \prime}_2},} & {{{\alpha \prime}_2} \rightarrow  {{\alpha \prime}_2},\quad \quad} & {{{\alpha \prime}_3} \rightarrow {{\alpha \prime}_3} + {{\alpha \prime}_2},} \\ {{{r\prime}_3}:} & {{{\alpha \prime}_1} \rightarrow {{\alpha \prime}_1},\quad \quad} & {{{\alpha \prime}_2} \rightarrow {{\alpha \prime}_2} + {{\alpha \prime}_3},\quad} & {{{\alpha \prime}_3} \rightarrow  {{\alpha \prime}_3},\quad} \\ \end{array}$$while the Weyl group \(\mathfrak{W}[A_2^{(2) +}]\) acts as
$$\begin{array}{*{20}c} {r\prime {\prime _1}:} & {\alpha \prime {\prime _1} \rightarrow  \alpha \prime {\prime _1},\;\;\quad} & {\alpha \prime {\prime _2} \rightarrow \alpha \prime {\prime _2} + 4\alpha \prime {\prime _1},\;} & {\alpha \prime {\prime _3} \rightarrow \alpha \prime {\prime _3},\;\;\quad \quad \;} \\ {r\prime {\prime _2}:} & {\alpha \prime {\prime _1} \rightarrow \alpha \prime {\prime _1} + \alpha \prime {\prime _2},} & {\alpha \prime {\prime _2} \rightarrow  \alpha \prime {\prime _2},\;\;\quad \quad} & {\alpha \prime {\prime _3} \rightarrow \alpha \prime {\prime _3} + \alpha \prime {\prime _2},\;} \\ {r\prime {\prime _3}:} & {\alpha \prime {\prime _1} \rightarrow \alpha \prime {\prime _1},\;\;\quad \;\;} & {\alpha \prime {\prime _2} \rightarrow \alpha \prime {\prime _2} + \alpha \prime {\prime _3},\;\;} & {\alpha \prime {\prime _3} \rightarrow  \alpha \prime {\prime _3}.\;\;\;\;\;\;\;} \\ \end{array}$$We see that the reflections coincide, \({\sigma _1} = r_1\prime = r_1{\prime \prime},\,{\sigma _2} = r_2\prime = r_2{\prime \prime},\,{\sigma _3} = r_3\prime = r_3{\prime \prime}\), as well as the scalar products, provided that we set \(2\alpha _1{\prime \prime} = \alpha _1\prime, \,\alpha _2{\prime \prime} = \alpha _2\prime, \,\alpha _3\prime = {\alpha _3}\) and \(\alpha _i\prime = \sqrt 2 {\alpha _i}\). The Coxeter group \(\mathfrak{B}\) generated by the reflections thus preserves the lattices
$$Q\prime = \sum\limits_i {{\mathbb Z}\alpha \prime _i}\qquad {\rm{and}}\qquad Q\prime \prime = \sum\limits_i {{\mathbb Z}\alpha \prime \prime _i},$$(4.61)showing explicitly that, in the present case, the lattices preserved by a Coxeter group are not unique — and might not even be dual to each other.
It follows, of course, that the Weyl groups of the KacMoody algebras \(A_1^{+ +}\) and \(A_1^{(2) +}\) are the same,
$$\mathfrak{W}[A_1^{+ +}] =\mathfrak{W} [A_2^{(2) +}] =\mathfrak{B} .$$(4.62) 
Consider now the Cartan matrix
$$A\prime \prime \prime = \left({\begin{array}{*{20}c} 2 & { 6} & 0 \\ { 1} & 2 & { 1} \\ 0 & { 1} & 2 \\\end{array}} \right),$$and its symmetrization
$$S\prime \prime \prime = \left({\begin{array}{*{20}c} {{1 \over 3}} & { 1} & 0 \\ { 1} & 2 & { 1} \\ 0 & { 1} & 2 \\ \end{array}} \right),$$The Weyl group \(\mathfrak{W}[A\prime \prime \prime ]\) of the corresponding KacMoody algebra is isomorphic to the Coxeter group \(\mathfrak{B}\) above since, according to the rules, the Coxeter exponents are identical. But the action is now
$$\begin{array}{*{20}c} {r\prime \prime {\prime _1}}: & {\alpha \prime \prime {\prime _1} \rightarrow  \alpha \prime \prime {\prime _1},\;\;\quad \;} & {\alpha \prime \prime {\prime _2} \rightarrow \alpha \prime \prime {\prime _2} + 6\alpha \prime \prime {\prime _1},} & {\alpha \prime \prime {\prime _3} \rightarrow \alpha \prime \prime {\prime _3}\quad \quad \quad} \\ {r\prime \prime {\prime _2}}: & {\alpha \prime \prime {\prime _1} \rightarrow \alpha \prime \prime {\prime _1} + \alpha \prime \prime {\prime _2},} & {\alpha \prime \prime {\prime _2} \rightarrow  \alpha \prime \prime {\prime _2}\quad \;\;\quad} & {\alpha \prime \prime {\prime _3} \rightarrow \alpha \prime \prime {\prime _3} + \alpha \prime \prime {\prime _2}\;} \\ {r\prime \prime {\prime _3}}:& {\alpha \prime \prime {\prime _1} \rightarrow \alpha \prime \prime {\prime _1},\;\quad \quad} & {\alpha \prime \prime {\prime _2} \rightarrow \alpha \prime \prime {\prime _2} + \alpha \prime \prime {\prime _3},} & {\alpha \prime \prime {\prime _3} \rightarrow  \alpha \prime \prime {\prime _3}\quad \quad} \\ \end{array}$$and cannot be made to coincide with the previous action by rescalings of the α′_{i}″’s. One can easily convince oneself of the inequivalence by computing the eigenvalues of the matrices S′, S″ and S′″ with respect to B.
4.8 Hyperbolic KacMoody algebras
Hyperbolic KacMoody algebras are by definition Lorentzian KacMoody algebras with the property that removing any node from their Dynkin diagram leaves one with a Dynkin diagram of affine or finite type. The Weyl group of hyperbolic KacMoody algebras is a crystallographic hyperbolic Coxeter group (as defined in Section 3.5). Conversely, any crystallographic hyperbolic Coxeter group is the Weyl group of at least one hyperbolic KacMoody algebra. Indeed, consider one of the lattices preserved by the Coxeter group as constructed in Section 3.6. The matrix with entries equal to the d_{ij} of that section is the Cartan matrix of a KacMoody algebra that has this given Coxeter group as Weyl group.
The hyperbolic KacMoody algebras have been classified in [154] and exist only up to rank 10 (see also [59]). In rank 10, there are four possibilities, known as \({E_{10}} \equiv E_8^{+ +},\,B{E_{10}} \equiv B_8^{+ +},\,D{E_{10}} \equiv D_8^{+ +}\) and \(C{E_{10}} \equiv A_{15}^{(2) +}\), BE_{10} and CE_{10} being dual to each other and possessing the same Weyl group (the notation will be explained below).
4.8.1 The fundamental domain \({\mathcal F}\)
For a hyperbolic KacMoody algebra, the fundamental weights Λ_{i} are timelike or null and lie within the (say) past lightcone. Similarly, the fundamental Weyl chamber \({\mathcal F}\) defined by \(\{v \in \mathcal{F} \Leftrightarrow (v \vert \alpha_i) \geq 0 \}\) also lies within the past lightcone and is a fundamental region for the action of the Weyl group on the Tits cone, which coincides in fact with the past light cone. All these properties carries over from our discussion of hyperbolic Coxeter groups in Section 3.
The positive imaginary roots α_{K} of the algebra fulfill (α_{K}Λ_{j}) ≥ 0 (with, for any K, strict inequality for at least one i) and hence, since they are nonspacelike, must lie in the future light cone. Recall indeed that the scalar product of two nonspacelike vectors with the same time orientation is nonpositive. For this reason, it is also of interest to consider the action of the Weyl group on the future lightcone, obtained from the action on the past lightcone by mere changes of signs. A fundamental region is clearly given by \( \mathcal{F}\). Any imaginary root is Weylconjugated to one that lies in \( \mathcal{F}\).
4.8.2 Roots and the root lattice
We have mentioned that not all points on the root lattice Q of a KacMoody algebras are actually roots. For hyperbolic algebras, there exists a simple criterion which enables one to determine whether a point on the root lattice is a root or not. We give it first in the case where all simple roots have equal length squared (assumed equal to two).
Theorem: Consider a hyperbolic KacMoody algebra such that (α_{i}α_{i}) = 2 for all simple roots α_{i}. Then, any point α on the root lattice Q with (αα) ≤ 2 is a root (note that (αα) is even). In particular, the set of real roots is the set of points on the root lattice with (αα) = 2, while the set of imaginary roots is the set of points on the root lattice (minus the origin) with (αα) ≤ 0. For a proof, see [116], Chapter 5.
The version of this theorem applicable to KacMoody algebras with different simple root lengths is the following.
Theorem: Consider a hyperbolic algebra with root lattice Q. Let a be the smallest length squared of the simple roots, a = min_{i}(α_{i}α_{i}). Then we have:
The set of all short real roots is {α ∈ Q (αα) = a}.
The set of all real roots is
$$\left\{{\alpha = \sum\limits_i {{k_i}} {\alpha _i} \in Q\,\vert \,(\alpha \vert \alpha) > 0\;{\rm{and}}\;{k_i}{{({\alpha _i}\vert {\alpha _i})} \over {(\alpha \vert \alpha)}} \in \;{\mathbb Z}\forall i} \right\}.$$The set of all imaginary roots is the set of points on the root lattice (minus the origin) with (αα) ≤ 0.
For a proof, we refer again to [116], Chapter 5.
We shall illustrate these theorems in the examples below. Note that it follows in particular from the theorems that if α is an imaginary root, all its integer multiples are also imaginary roots.
4.8.3 Examples
We discuss here briefly two examples, namely \(A_1^{+ +}\), for which all simple roots have equal length, and \(A_2^{(2) +}\), with respective Dynkin diagrams shown in Figures 17 and 18.
4.8.3.1 The KacMoody Algebra \(A_1^{+ +}\)
This is the algebra associated with vacuum fourdimensional Einstein gravity and the BKL billiard. We encountered its Weyl group PGL(2, ℤ) already in Section 3.1.1. The algebra is also denoted AE_{3}, or H_{3}. The Cartan matrix is
As it follows from our analysis in Section 3.1.1, the simple roots may be identified with the following linear forms α_{i}(β) in the threedimensional space of the β^{i}’s,
with scalar product
for two linear forms F = F_{i}β^{i} and G = G_{i}β^{i}. It is sometimes convenient to analyze the root system in terms of an “affine” level ℓ that counts the number of times the root α_{3} occurs: The root kα_{1} + mα_{2} + ℓα_{3} has by definition level ℓ^{Footnote 13}. We shall consider here only positive roots for which k, m, ℓ ≥ 0.
Applying the first theorem, one easily verifies that the only positive roots at level zero are the roots kα_{1} + mα_{2}, k − m ≤ 1 (k, m ≥ 0) of the affine subalgebra \(A_1^ +\). When k = m, the root is imaginary and has length squared equal to zero. When k − m = 1, the root is real and has length squared equal to two.
Similarly, the only roots at level one are (m + a)α_{1} + mα_{2} + α_{3} with a^{2} ≤ m, i.e., \( \left[ {\sqrt m} \right] \leq a \leq \left[ {\sqrt m} \right]\). Whenever \(\sqrt m\) is an integer, the roots \(\left({m \pm \sqrt m} \right){\alpha _1} + m{\alpha _2} + {\alpha _3}\) have squared length equal to two and are real. The roots (m + a)α_{1} + mα_{2} + α_{3} with a^{2} < m are imaginary and have squared length equal to 2(a^{2} + 1 − m) ≤ 0. In particular, the root m(α_{1} + α_{2}) + α_{3} has length squared equal to 2(1 − m). Of all the roots at level one with m > 1, these are the only ones that are in the fundamental domain \(\mathcal{F}\) (i.e., that fulfill (βα_{i}) ≤ 0). When m = 1, none of the level1 roots is in \(\mathcal{F}\) and is either in the Weyl orbit of α_{1} + α_{2}, or in the Weyl orbit of α_{3}.
We leave it to the reader to verify that the roots at level two that are in the fundamental domain \(\mathcal{F}\) take the form (m − 1)α_{1} + mα_{2} + 2α_{3} and m(α_{1} + α_{2}) + 2α_{3} with m ≥ 4. Further information on the roots of \(A_1^{+ +}\) may be found in [116], Chapter 11, page 215.
4.8.3.2 The KacMoody Algebra \(A_2^{(2) +}\)
This is the algebra associated with the EinsteinMaxwell theory (see Section 7). The notation will be explained in Section 4.9. The Cartan matrix is
and there are now two lengths for the simple roots. The scalar products are
One may realize the simple roots as the linear forms
in the threedimensional space of the β^{i}’s with scalar product Equation (4.65).
The real roots, which are Weyl conjugate to one of the simple roots α_{1} or α_{2} (α_{3} is in the same Weyl orbit as α_{2}), divide into long and short real roots. The long real roots are the vectors on the root lattice with squared length equal to two that fulfill the extra condition in the theorem. This condition expresses here that the coefficient of α_{1} should be a multiple of 4. The short real roots are the vectors on the root lattice with length squared equal to onehalf. The imaginary roots are all the vectors on the root lattice with length squared ≤ 0.
We define again the level ℓ as counting the number of times the root α_{3} occurs. The positive roots at level zero are the positive roots of the twisted affine algebra \(A_2^{(2)}\), namely, α_{1} and (2m + a)α_{1}+mα_{2}, m = 1, 2, 3, ⋯, with a = −2, −1, 0, 1, 2 for m odd and a = −1, 0, 1 for m odd. Although belonging to the root lattice and of length squared equal to two, the vectors (2m ± 2)α_{1} + mα_{2} are not long real roots when m is even because they fail to satisfy the condition that the coefficient (2m ± 2) of α_{1} is a multiple of 4. The roots at level zero are all real, except when a = 0, in which case the roots m(2α_{1} + α_{2}) have zero norm.
To get the long real roots at level one, we first determine the vectors α = α_{3} + kα_{1} + mα_{2} of squared length equal to two. The condition (αα) = 2 easily leads to m = p^{2} for some integer p ≥ 0 and k = 2p^{2} ± 2p = 2p(p ± 1). Since k is automatically a multiple of 4 for all p = 0, 1, 2, 3, ⋯, the corresponding vectors are all long real roots. Similarly, the short real roots at level one are found to be (2p^{2} + 1)α_{1} + (p^{2} + p + 1)α_{2} + α_{3} and (2p^{2} + 4p + 3)α_{1} + (p^{2} + p + 1)α_{2} + α_{3} for p a nonnegative integer.
Finally, the imaginary roots at level one in the fundamental domain \(\mathcal{F}\) read (2m − 1)α_{1} + mα_{2} + α_{3} and 2mα_{1} + mα_{2} + α_{3} where m is an integer greater than or equal to 2. The first roots have length squared equal to \( 2m + {5 \over 2}\), the second have length squared equal to −2m + 2.
4.9 Overextensions of finitedimensional Lie algebras
An interesting class of Lorentzian KacMoody algebras can be constructed by adding simple roots to finitedimensional simple Lie algebras in a particular way which will be described below. These are called “overextensions”.
In this section, we let \(\mathfrak{g}\) be a complex, finitedimensional, simple Lie algebra of rank r, with simple roots α_{1}, ⋯, α_{r}. As stated above, normalize the roots so that the long roots have length squared equal to 2 (the short roots, if any, have then length squared equal to 1 (or 2/3 for G_{2})). The roots of simplylaced algebras are regarded as long roots.
Let α = ∑_{i}n_{i}α_{i}, n_{i} ≥ 0 be a positive root. One defines the height of α as
Among the roots of \(\mathfrak{g}\), there is a unique one that has highest height, called the highest root. We denote it by θ. It is long and it fulfills the property that (θα_{i}) ≥ 0 for all simple roots α_{i}, and
(see, e.g., [85]). We denote by V the rdimensional Euclidean vector space spanned by α_{i} (i = 1, ⋯, r). Let M_{2} be the twodimensional Minkowski space with basis vectors u and v so that (uu) = (vv) = 0 and (uv) = 1. The metric in the space V ⊕ M_{2} has clearly Minkowskian signature (−, +, +, ⋯, +) so that any KacMoody algebra whose simple roots span V ⊕ M_{2} is necessarily Lorentzian.
4.9.1 Untwisted overextensions
The standard overextensions \(\mathfrak{g}^{++}\) are obtained by adding to the original roots of \(\mathfrak{g}\) the roots \(\alpha _i\prime\) The matrix \({A_{ij}} = 2{{\left({{\alpha _i}\vert {\alpha _j}} \right)} \over {\left({{\alpha _i}\vert {\alpha _i}} \right)}}\) where i, j = −1, 0, 1, ⋯, r is a (generalized) Cartan matrix and defines indeed a KacMoody algebra.
The root α_{0} is called the affine root and the algebra \(\mathfrak{g}^+\) (\(\mathfrak{g}^{(1)}\) in Kac’s notations [116]) with roots α_{0}, α_{1}, ⋯, α_{r} is the untwisted affine extension of \(\mathfrak{g}\). The root α_{−1} is known as the overextended root. One clearly has \({\rm{rank}}({\mathfrak{g}^{+ +}}) = {\rm{rank}}(\mathfrak{g}) + 2\). The overextended root has vanishing scalar product with all other simple roots except α_{0}. One has explicitly (α_{−1} α_{−1}) = 2 = (α_{0}α_{0}) and (α_{−1} α_{0}) = −1, which shows that the overextended root is attached to the affine root (and only to the affine root) with a single link.
Of these Lorentzian algebras, the following ones are hyperbolic:
\(A_k^{+ +}(k \leq 7)\),
\(B_k^{+ +}(k \leq 8)\),
\(C_k^{+ +}(k \leq 4)\),
\(D_k^{+ +}(k \leq 8)\),
\(G_2^{+ +}\),
\(F_4^{+ +}\),
\(E_k^{+ +}(k = 6,\,7,\,8)\).
The algebras \(B_8^{+ +},\,D_8^{+ +}\) and \(E_8^{+ +}\) are also denoted BE_{10}, DE_{10} and E_{10}, respectively.
4.9.1.1 A special property of E_{10}
Of these maximal rank hyperbolic algebras, E_{10} plays a very special role. Indeed, one can verify that the determinant of its Cartan matrix is equal to −1. It follows that the lattice of E_{10} is selfdual, i.e., that the fundamental weights belong to the root lattice of E_{10}. In view of the above theorem on roots of hyperbolic algebras and of the hyperboliticity of E_{10}, the fundamental weights of E_{10} are actually (imaginary) roots since they are nonspacelike. The root lattice of E_{10} is the only Lorentzian, even, selfdual lattice in 10 dimensions (these lattices exist only in 2 mod 8 dimensions).
4.9.2 Root systems in Euclidean space
In order to describe the “twisted” overextensions, we need to introduce the concept of a “root system”.
A root system in a real Euclidean space V is by definition a finite subset Δ of nonzero elements of V obeying the following two conditions:
The elements of Δ are called the roots. From the definition one can prove the following properties [93]:
 1.
If α ∈ Δ, then −α ∈ Δ.
 2.
If α ∈ Δ, then the only elements of Δ proportional to α are ±½α, ±α, ±2α. If only ±α occurs (for all roots α), the root system is said to be reduced (proper in “Araki terminology” [5]).
 3.
If α, β ∈ Δ, then 0 ≤ A_{α,β}A_{β,α} ≤ 4, i.e., A_{α,β} = 0, ±1, ±2, ±3, ±4; the last occurrence appearing only for β = ±2α, i.e., for nonreduced systems. (The proof of this point requires the use of the Schwarz inequality.)
 4.
If α, β ∈ Δ are not proportional to each other and (αα) ≤ (ββ) then A_{α,β} = 0, ±1. Moreover if (αβ) = 0, then (ββ) = (αα), 2 (αα), or 3 (αα).
 5.
If If α, β ∈ Δ, but α — β ∉ Δ ∪ 0, then (αβ) ≤ 0 and, as a consequence, if α, β ∈ Δ but α ± β ∉ Δ ∪ 0 then (αβ) = 0. That (αβ) ≤ 0 can be seen as follows. Clearly, α and β can be assumed to be linearly independent^{Footnote 14}. Now, assume (αβ) > 0. By the previous point, A_{α,β} = 1 or A_{β,α} = 1. But then either α − A_{α,β}β = α − β ∈ Δ or −(β − A_{β,α}α) = α − β ∈ Δ by (4.72), contrary to the assumption. This proves that (αβ) ≤ 0.
Since Δ spans the vector space V, one can chose a basis {α_{i}} of elements of V within Δ. This can furthermore be achieved in such a way the α_{i} enjoy the standard properties of simple roots of Lie algebras so that in particular the concepts of positive, negative and highest roots can be introduced [93].
All the abstract root systems in Euclidean space have been classified (see, e.g., [93]) with the following results:
The most general root system is obtained by taking a union of irreducible root systems. An irreducible root system is one that cannot be decomposed into two disjoint nonempty orthogonal subsets.
The irreducible reduced root systems are simply the root systems of finitedimensional simple Lie algebras (A_{n} with n ≥ 1, B_{n} with n ≥ 3, C_{n} with n ≥ 2, D_{n} with n ≥ 4, G_{2}, F_{4}, E_{6}, E_{7} and E_{8}).
Irreducible nonreduced root systems are all given by the socalled (BC)_{n}systems. A (BC)_{n}system is obtained by combining the root system of the algebra B_{n} with the root system of the algebra C_{n} in such a way that the long roots of B_{n} are the short roots of C_{n}. There are in that case three different root lengths. Explicitly Δ is given by the n unit vectors \({\vec e_k}\) and their opposite \( {\vec e_k}\) along the Cartesian axis of an ndimensional Euclidean space, the 2n vectors \(\pm 2{\vec e_k}\) obtained by multiplying the previous vectors by 2 and the 2n(n − 1) diagonal vectors \(\pm {\vec e_k} \pm {\vec e_{k\prime}}\), with k ≠ k′ and k, k′ = 1, …, n. The n = 3 case is pictured in Figure 19. The Dynkin diagram of (BC)_{r} is the Dynkin diagram of B_{r} with a double circle ⊚ over the simple short root, say α_{1}, to indicate that 2α_{1} is also a root.
It is sometimes convenient to rescale the roots by the factor \((1/\sqrt 2)\) so that the highest root θ = 2(α_{1} + α_{2} + ⋯ + α_{r}) [93] of the (BC)system has length 2 instead of 4.
4.9.3 Twisted overextensions
We follow closely [95]. Twisted affine algebras are related to either the (BC)root systems or to extensions by the highest short root (see [116], Proposition 6.4).
4.9.3.1 Twisted overextensions associated with the (BC)root systems
These are the overextensions relevant for some of the gravitational billiards. The construction proceeds as for the untwisted overextensions, but the starting point is now the (BC)_{r} root system with rescaled roots. The highest root has length squared equal to 2 and has nonvanishing scalar product only with α_{r} ((α_{r}θ) = 1). The overextension procedure (defined by the same formulas as in the untwisted case) yields the algebra \((BC)_r^{+ +}\), also denoted \(A_{2r}^{(2) +}\).
There is an alternative overextension \(A_{2r}^{(2)\prime +}\) that can be defined by starting this time with the algebra C_{r} but taking onehalf the highest root of C_{r} to make the extension (see [116], formula in Paragraph 6.4, bottom of page 84). The formulas for α_{0} and α_{−1} are 2α_{0} = u−θ and 2α_{−1} = −u−v (where θ is now the highest root of C_{r}). The Dynkin diagram of \(A_{2r}^{(2)\prime +}\) is dual to that of \(A_{2r}^{(2) +}\). (Duality amounts to reversing the arrows in the Dynkin diagram, i.e., replacing the (generalized) Cartan matrix by its transpose.)
The algebras \(A_{2r}^{(2) +}\) and \(A_{2r}^{(2)\prime +}\) have rank r + 2 and are hyperbolic for r ≤ 4. The intermediate affine algebras are in all cases the twisted affine algebras \(A_{2r}^{(2)}\). We shall see in Section 7 that by coupling to threedimensional gravity a coset model \({\mathcal G}/{\mathcal K}({\mathcal G})\), where the socalled restricted root system (see Section 6) of the (real) Lie algebra \(\mathfrak{g}\) of the Lie group \({\mathcal G}\) is of (BC)_{r}type, one can realize all the \(A_{2r}^{(2) +}\) algebras.
4.9.3.2 Twisted overextensions associated with the highest short root
We denote by θ_{s} the unique short root of heighest weight. It exists only for nonsimply laced algebras and has length 1 (or 2/3 for G_{2}). The twisted overextensions are defined as the standard overextensions but one uses instead the highest short root θ_{s}. The formulas for the affine and overextended roots are
or
(We choose the overextended root to have the same length as the affine root and to be attached to it with a single link. This choice is motivated by considerations of simplicity and yields the fourth rank ten hyperbolic algebra when \(\mathfrak{g}={C_8}\).)
The affine extensions generated by α_{0}, ⋯, α_{r} are respectively the twisted affine algebras \(D_{r + 1}^{(2)}\,(\mathfrak{g} = {B_r}),\,A_{2r  1}^{(2)}\,(\mathfrak{g} = {C_r}),\,E_6^{(2)}\,(\mathfrak{g} = {F_4})\) and \(D_4^{(3)}\,(\mathfrak{g} = {G_2})\). These twisted affine algebras are related to external automorphisms of D_{r+1}, A_{2r−1}, E_{6} and D_{4}, respectively (the same holds for \(A_{2r}^{(2)}\) above) [116]. The corresponding twisted overextensions have the following features.
The overextensions \(D_{r + 1}^{(2) +}\) have rank r + 2 and are hyperbolic for r ≤ 4.
The overextensions \(A_{2r  1}^{(2) +}\) have rank r + 2 and are hyperbolic for r ≤ 8. The last hyperbolic case, r = 8, yields the algebra \(A_{15}^{(2) +}\), also denoted CE_{10}. It is the fourth rank10 hyperbolic algebra, besides E_{10}, BE_{10} and DE_{10}.
The overextensions \(E_6^{(2) +}\) (rank 6) and \(D_4^{(3) +}\) (rank 4) are hyperbolic.
We list in Table 14 the Dynkin diagrams of all twisted overextensions.
A satisfactory feature of the class of overextensions (standard and twisted) is that it is closed under duality. For instance, \(A_{2r  1}^{(2) +}\) is dual to \(B_r^{+ +}\). In fact, one could get the twisted overextensions associated with the highest short root from the standard overextensions precisely by requiring closure under duality. A similar feature already holds for the affine algebras.
Note also that while not all hyperbolic KacMoody algebras are symmetrizable, the ones that are obtained through the process of overextension are.
4.9.4 Algebras of GaberdielOliveWest type
One can further extend the overextended algebras to get “triple extensions” or “very extended algebras”. This is done by adding a further simple root attached with a single link to the overextended root of Section 4.9. For instance, in the case of E_{10}, one gets E_{11} with the Dynkin diagram displayed in Figure 20. These algebras are Lorentzian, but not hyperbolic.
The very extended algebras belong to a more general class of algebras considered by Gaberdiel, Olive and West in [86]. These are defined to be algebras with a connected Dynkin diagram that possesses at least one node whose deletion yields a diagram with connected components that are of finite type except for at most one of affine type. For a hyperbolic algebra, the deletion of any node should fulfill this condition. The algebras of Gaberdiel, Olive and West are Lorentzian if not of finite or affine type [153, 86]. They include the overextensions of Section 4.9. The untwisted and twisted very extended algebras are clearly also of this type, since removing the affine root gives a diagram with the requested properties.
Higher order extensions with special additional properties have been investigated in [78].
4.10 Regular subalgebras of KacMoody algebras
This section is based on [96].
4.10.1 Definitions
Let \({\mathfrak g}\) be a KacMoody algebra, and let \({\bar {\mathfrak g}}\) be a subalgebra of \({\mathfrak g}\) with triangular decomposition \({\bar {\mathfrak g}} = {{\bar {\mathfrak n}}_ } \oplus {\bar {\mathfrak h}} \oplus {{\bar {\mathfrak n}}_ +}\). We assume that \({\bar {\mathfrak g}}\) is canonically embedded in \({\mathfrak g}\), i.e., that the Cartan subalgebra \({\bar {\mathfrak h}}\) of \({\bar {\mathfrak g}}\) is a subalgebra of the Cartan subalgebra \({\mathfrak h}\) of \({\mathfrak g}\), \({\bar {\mathfrak h}} \subset {\mathfrak h}\), so that \({\bar {\mathfrak h}} = {\bar {\mathfrak g}} \cap {\mathfrak h}\). We shall say that \({\bar {\mathfrak g}}\) is regularly embedded in \({\mathfrak g}\) (and call it a “regular subalgebra”) if and only if two conditions are fulfilled: (i) The root generators of \({\bar {\mathfrak g}}\) are root generators of \({\mathfrak g}\), and (ii) the simple roots of \({\bar {\mathfrak g}}\) are real roots of \({\mathfrak g}\). It follows that the Weyl group of \({\bar {\mathfrak g}}\) is a subgroup of the Weyl group of \({\mathfrak g}\) and that the root lattice of \({\bar {\mathfrak g}}\) is a sublattice of the root lattice of \({\mathfrak g}\).
The second condition is automatic in the finitedimensional case where there are only real roots. It must be separately imposed in the general case. Consider for instance the rank 2 KacMoody algebra \({\mathfrak g}\) with Cartan matrix
Let
It is easy to verify that x, y, z define an A_{1} subalgebra of \({\mathfrak g}\) since [z, x] = 2x, [z, y] = −2y and [x, y] = z. Moreover, the Cartan subalgebra of A_{1} is a subalgebra of the Cartan subalgebra of g, and the step operators of A_{1} are step operators of \({\mathfrak g}\). However, the simple root α = α_{1} + α_{2} of A_{1} (which is an A_{1}real root since A_{1} is finitedimensional), is an imaginary root of \({\mathfrak {g}}: {\alpha _1} + {\alpha _2}\) has norm squared equal to −2. Even though the root lattice of A_{1} (namely, {±α}) is a sublattice of the root lattice of \({\mathfrak g}\), the reflection in α is not a Weyl reflection of \({\mathfrak g}\). According to our definition, this embedding of A_{1} in \({\mathfrak g}\) is not a regular embedding.
4.10.2 Examples — Regular subalgebras of E_{10}
We shall describe some regular subalgebras of E_{10}. The Dynkin diagram of E_{10} is displayed in Figure 21.
4.10.2.1 \({A_9} \subset {\mathcal B} \subset{E_{10}}\)
A first, simple, example of a regular embedding is the embedding of A_{9} in E_{10} which will be used to define the level when trying to reformulate elevendimensional supergravity as a nonlinear sigma model. This is not a maximal embedding since one can find a proper subalgebra \({\mathcal B}\) of E_{10} that contains A_{9}. One may take for \({\mathcal B}\) the KacMoody subalgebra of E_{10} generated by the operators at levels 0 and ±2, which is a subalgebra of the algebra containing all operators of even level^{Footnote 15}. It is regularly embedded in E_{10}. Its Dynkin diagram is shown in Figure 22.
In terms of the simple roots of E_{10}, the simple roots of \({\mathcal B}\) are α_{1} through α_{9} and ᾱ_{10} = 2α_{10} + α_{1} + 2α_{2} + 3α_{3} + 2α_{4} + α_{5}. The algebra \({\mathcal B}\) is Lorentzian but not hyperbolic. It can be identified with the “very extended” algebra \(E_7^{+ + +}\) [86].
4.10.2.2 DE_{10} ⊂ E_{10}
In [67], Dynkin has given a method for finding all maximal regular subalgebras of finitedimensional simple Lie algebras. The method is based on using the highest root and is not generalizable as such to general KacMoody algebras for which there is no highest root. Nevertherless, it is useful for constructing regular embeddings of overextensions of finitedimensional simple Lie algebras. We illustrate this point in the case of E_{8} and its overextension \({E_{10}} \equiv E_8^{+ +}\). In the notation of Figure 21, the simple roots of E_{8} (which is regularly embedded in E_{10}) are α_{1}, ⋯, α_{7} and α_{10}.
Applying Dynkin’s procedure to E_{8}, one easily finds that D_{8} can be regularly embedded in E_{8}. The simple roots of D_{8} ⊂ E_{8} are α_{2}, α_{3}, α_{4}, α_{5}, α_{6}, α_{7}, α_{10} and \(\beta \equiv  {\theta _{{E_8}}}\), where
is the highest root of E_{8}. One can replace this embedding, in which a simple root of D_{8}, namely β, is a negative root of E_{8} (and the corresponding raising operator of D_{8} is a lowering operator for E_{8}), by an equivalent one in which all simple roots of D_{8} are positive roots of E_{8}.
This is done as follows. It is reasonable to guess that the searchedfor Weyl element that maps the “old” D_{8} on the “new” D_{8} is some product of the Weyl reflections in the four E_{8}roots orthogonal to the simple roots α_{3}, α_{4}, α_{5}, α_{6} and α_{7}, expected to be shared (as simple roots) by E_{8}, the old D_{8} and the new D_{8} — and therefore to be invariant under the searchedfor Weyl element. This guess turns out to be correct: Under the action of the product of the commuting E_{8}Weyl reflections in the E_{8}roots μ_{1} = 2α_{1} + 3α_{2} + 5α_{3} + 4α_{4} + 3α_{5} + 2α_{6} + α_{7} + 3α_{10} and μ_{2} = 2α_{1}+4α_{2} +5α_{3}+4α_{4}+3α_{5} + 2α_{6}+α_{7}+2α_{10}, the set of D_{8}roots {α_{2}, α_{3}, α_{4}, α_{5}, α_{6}, α_{7}, α_{10}, β} is mapped on the equivalent set of positive roots {α_{10}, α_{3}, α_{4}, α_{5}, α_{6}, α_{7}, α_{2}, β}, where
In this equivalent embedding, all raising operators of D_{8} are also raising operators of E_{8}. What is more, the highest root of D_{8},
is equal to the highest root of E_{8}. Because of this, the affine root α_{8} of the untwisted affine extension \(E_8^{+}\) can be identified with the affine root of \(D_8^{+}\), and the overextended root α_{9} can also be taken to be the same. Hence, DE_{10} can be regularly embedded in E_{10} (see Figure 23).
The embedding just described is in fact relevant to string theory and has been discussed from various points of view in previous papers [125, 23]. By dimensional reduction of the bosonic sector of elevendimensional supergravity on a circle, one gets, after dropping the KaluzaKlein vector and the 3form, the bosonic sector of pure \({\mathcal N} = 1\) tendimensional supergravity. The simple roots of DE_{10} are the symmetry walls and the electric and magnetic walls of the 2form and coincide with the positive roots given above [45]. A similar construction shows that \(A_8^{+ +}\) can be regularly embedded in E_{10}, and that DE_{10} can be regularly embedded in \(B_8^{+ +}\). See [106] for a recent discussion of DE_{10} in the context of Type I supergravity.
4.10.3 Further properties
As we have just seen, the raising operators of \({\bar {\mathfrak g}}\) might be raising or lowering operators of \({\mathfrak g}\). We shall consider here only the case when the positive (respectively, negative) root generators of \({\bar {\mathfrak g}}\) are also positive (respectively, negative) root generators of \({\mathfrak g}\), so that \({{\bar {\mathfrak n}}_ } = {{\mathfrak n}_ } \cap {\bar {\mathfrak g}}\) and \({{\bar {\mathfrak n}}_ +} = {{\mathfrak n}_ +} \cap {\bar {\mathfrak g}}\) (“positive regular embeddings”). This will always be assumed from now on.
In the finitedimensional case, there is a useful criterion to determine regular algebras from subsets of roots. This criterion, which does not use the highest root, has been generalized to KacMoody algebras in [76]. It covers also nonmaximal regular subalgebras and goes as follows:
Theorem: Let \(\Phi _{{\rm{real}}}^ +\) be the set of positive real roots of a KacMoody algebra \({\mathfrak g}\). Let \({\gamma _1},\, \ldots, \,{\gamma _n} \in \Phi _{{\rm{real}}}^ +\) be chosen such that none of the differences γ_{i} − γ_{j} is a root of \({\mathfrak g}\). Assume furthermore that the γ_{i}’s are such that the matrix C = [C_{ij}] = [2 (γ_{i}γ_{j}) / (γ_{i}γ_{i})] has nonvanishing determinant. For each 1 ≤ i ≤ n, choose nonzero root vectors E_{i} and F_{i} in the onedimensional root spaces corresponding to the positive real roots γ_{i} and the negative real roots −γ_{i}, respectively, and let H_{i} = [E_{i}, F_{i}] be the corresponding element in the Cartan subalgebra of \({\mathfrak g}\). Then, the (regular) subalgebra of \({\mathfrak g}\) generated by {E_{i}, F_{i}, H_{i}}, i = 1, ⋯, n, is a KacMoody algebra with Cartan matrix [C_{ij}].
Proof: The proof of this theorem is given in [76]. Note that the Cartan integers \(2{{({\gamma _i}\vert {\gamma _j})} \over {({\gamma _i}\vert {\gamma _i})}}\) are indeed integers (because the γ_{i}’s are positive real roots), which are nonpositive (because γ_{i} − γ_{j} is not a root), so that [C_{ij}] is a Cartan matrix.
4.10.3.1 Comments
 1.
When the Cartan matrix is degenerate, the corresponding KacMoody algebra has nontrivial ideals [116]. Verifying that the ChevalleySerre relations are fulfilled is not sufficient to guarantee that one gets the KacMoody algebra corresponding to the Cartan matrix [C_{ij}] since there might be nontrivial quotients. Situations in which the algebra generated by the set {E_{i}, F_{i}, H_{i}} is the quotient of the KacMoody algebra with Cartan matrix [C_{ij}] by a nontrivial ideal were discussed in [96].
 2.
If the matrix [C_{ij}] is decomposable, say C = D ⊕ E with D and E indecomposable, then the KacMoody algebra \({\mathbb K}{\mathbb M}(C)\) generated by C is the direct sum of the KacMoody algebra \({\mathbb K}{\mathbb M}(D)\) generated by D and the KacMoody algebra \({\mathbb K}{\mathbb M}(E)\) generated by E. The subalgebras \({\mathbb K}{\mathbb M}(D)\) and \({\mathbb K}{\mathbb M}(E)\) are ideals. If C has nonvanishing determinant, then both D and E have nonvanishing determinant. Accordingly, \({\mathbb K}{\mathbb M}(D)\) and \({\mathbb K}{\mathbb M}(E)\) are simple [116] and hence, either occur faithfully or trivially. Because the generators E_{i} are linearly independent, both \({\mathbb K}{\mathbb M}(D)\) and \({\mathbb K}{\mathbb M}(E)\) occur faithfully. Therefore, in the above theorem the only case that requires special treatment is when the Cartan matrix C has vanishing determinant.
As we have mentioned above, it is convenient to universally normalize the Killing form of KacMoody algebras in such a way that the long real roots have always the same squared length, conveniently taken equal to two. It is then easily seen that the Killing form of any regular KacMoody subalgebra of E_{10} coincides with the invariant form induced from the Killing form of E_{10} through the embedding since E_{10} is “simply laced”. This property does not hold for nonregular embeddings as the example given in Section 4.1 shows: The subalgebra A_{1} considered there has an induced form equal to minus the standard Killing form.
5 KacMoody Billiards I — The Case of Split Real Forms
In this section we will begin to explore in more detail the correspondence between Lorentzian Coxeter groups and the limiting behavior of the dynamics of gravitational theories close to a spacelike singularity.
We have seen in Section 2 that in the BKLlimit, the dynamics of gravitational theories is equivalent to a billiard dynamics in a region of hyperbolic space. In the generic case, the billiard region has no particular feature. However, we have seen in Section 3 that in the case of pure gravity in four spacetime dimensions, the billiard region has the remarkable property of being the fundamental domain of the Coxeter group PGL(2, ℤ) acting on twodimensional hyperbolic space.
This is not an accident. Indeed, this feature arises for all gravitational theories whose toroidal dimensional reduction to three dimensions exhibits hidden symmetries, in the sense that the reduced theory can be reformulated as threedimensional gravity coupled to a nonlinear sigmamodel based on \({{\mathcal U}_3}/{\mathcal K}({{\mathcal U}_3})\), where \({\mathcal K}({{\mathcal U}_3})\) is the maximal compact subgroup of \({{\mathcal U}_3}\). The “hidden” symmetry group \({{\mathcal U}_3}\) is also called, by a generalization of language, “the Uduality group” [142]. This situation covers the cases of pure gravity in any spacetime dimension, as well as all known supergravity models. In all these cases, the billiard region is the fundamental domain of a Lorentzian Coxeter group (“Coxeter billiard”). Furthermore, the Coxeter group in question is crystallographic and turns out to be the Weyl group of a Lorentzian KacMoody algebra. The billiard table is then the fundamental Weyl chamber of a Lorentzian KacMoody algebra [45, 46] and the billiard is also called a “KacMoody billiard”. This enables one to reformulate the dynamics as a motion in the Cartan subalgebra of the Lorentzian KacMoody algebra, hinting at the potential — and still conjectural at this stage — existence of a deeper, infinitedimensional symmetry of the theory.
The purpose of this section is threefold:
 1.
First, we exhibit other theories besides pure gravity in four dimensions which also lead to a Coxeter billiard. We stress further how exceptional these theories are in the space of all theories described by the action Equation (2.1).
 2.
Second, we show how to reformulate the dynamics as a motion in the Cartan subalgebra of a Lorentzian KacMoody algebra.
 3.
Finally, we connect the Lorentzian KacMoody algebra that appears in the BKLlimit to the “hidden” symmetry group \({{\mathcal U}_3}\) in the simplest case when the real Lie algebra \({{\mathfrak u}_3}\) of the group \({{\mathcal U}_3}\) is the split real form of the corresponding complexified Lie algebra \({\mathfrak u}_3^{\mathbb C}\). (These concepts will be defined below.) The general case will be dealt with in Section 7, after we have recalled the most salient features of the theory of real forms in Section 6.
5.1 More on Coxeter billiards
5.1.1 The Coxeter billiard of pure gravity in D spacetime dimensions
We start by providing other examples of theories leading to regular billiards, focusing first on pure gravity in any number of D (> 3) spacetime dimensions. In this case, there are d = D − 1 scale factors β^{i} and the relevant walls are the symmetry walls, Equation (2.48),
and the curvature wall, Equation (2.49),
There are thus d relevant walls, which define a simplex in (d − 1)dimensional hyperbolic space The scalar products of the linear forms defining these walls are easily computed. One finds as nonvanishing products
The matrix of the scalar products of the wall forms is thus the Cartan matrix of the (simplylaced) Lorentzian KacMoody algebra \(A_{d  2}^{+ +}\) with Dynkin diagram as in Figure 24. The roots of the underlying finitedimensional algebra A_{d−2} are given by s_{i} (i = 1, ⋯, d − 3) and r. The affine root is s_{d−2} and the overextended root is s_{d−1}.
Accordingly, in the case of pure gravity in any number of spacetime dimensions, one finds also that the billiard region is regular. This provides new examples of Coxeter billiards, with Coxeter groups \(A_{d  2}^{+ +}\), which are also KacMoody billiards since the Coxeter groups are the Weyl groups of the KacMoody algebras \(A_{d  2}^{+ +}\).
5.1.2 The Coxeter billiard for the coupled gravity3Form system Coxeter polyhedra
Let us review the conditions that must be fulfilled in order to get a KacMoody billiard and let us emphasize how restrictive these conditions are. The billiard region computed from any theory coupled to gravity with n dilatons in D = d + 1 dimensions always defines a convex polyhedron in a (d + n − 1)dimensional hyperbolic space \({{\mathcal H}_{d1}}\). In the general case, the dihedral angles between adjacent faces of \({{\mathcal H}_{d+n1}}\) can take arbitrary continuous values, which depend on the dilaton couplings, the spacetime dimensions and the ranks of the pforms involved. However, only if the dihedral angles are integer submultiples of π (meaning of the form π/k for k ∈ ℤ_{≥2}) do the reflections in the faces of \({{\mathcal H}_{d+n1}}\) define a Coxeter group. In this special case the polyhedron is called a Coxeter polyhedron. This Coxeter group is then a (discrete) subgroup of the isometry group of \({{\mathcal H}_{d+n1}}\).
In order for the billiard region to be identifiable with the fundamental Weyl chamber of a KacMoody algebra, the Coxeter polyhedron should be a simplex, i.e., bounded by d + n walls in a d + n − 1dimensional space. In general, the Coxeter polyhedron need not be a simplex.
There is one additional condition. The angle ϑ between two adjacent faces i and j is given, in terms of the Coxeter exponents, by
Coxeter groups that correspond to Weyl groups of KacMoody algebras are the crystallographic Coxeter groups for which m_{ij} ∈ {2, 3, 4, 6, ∞}. So, the requirement for a gravitational theory to have a KacMoody algebraic description is not just that the billiard region is a Coxeter simplex but also that the angles between adjacent walls are such that the group of reflections in these walls is crystallographic.
These conditions are very restrictive and hence gravitational theories which can be mapped to a KacMoody algebra in the BKLlimit are rare.
5.1.2.1 The Coxeter billiard of elevendimensional supergravity
Consider for instance the action (2.1) for gravity coupled to a single threeform in D = d + 1 spacetime dimensions. We assume D ≥ 6 since in lower dimensions the 3form is equivalent to a scalar (D = 5) or has no degree of freedom (D < 5).
Theorem: Whenever a pform (p ≥ 1) is present, the curvature wall is subdominant as it can be expressed as a linear combination with positive coefficients of the electric and magnetic walls of the pforms. (These walls are all listed in Section 2.5.)
Proof: The dominant electric wall is (assuming the presence of a dilaton)
while one of the magnetic wall reads
so that the dominant curvature wall is just the sum e_{1⋯p}(β) + m_{1,p+1, ⋯, d−2}(β).
It follows that in the case of gravity coupled to a single threeform in D = d + 1 spacetime dimensions, the relevant walls are the symmetry walls, Equation (2.48),
(as always) and the electric wall
(D ≥ 8) or the magnetic wall
(D ≤ 8). Indeed, one can express the magnetic walls as linear combinations with (in general noninteger) positive coefficients of the electric walls for D ≥ 8 and vice versa for D ≤ 8. Hence the billiard table is always a simplex (this would not be true had one a dilaton and various forms with different dilaton couplings).
However, it is only for D = 11 that the billiard is a Coxeter billiard. In all the other spacetime dimensions, the angle between the relevant pform wall and the symmetry wall that does not intersect it orthogonally is not an integer submultiple of π. More precisely, the angle between
the magnetic wall β^{1} and the symmetry wall β^{2} − β^{1} (D = 6),
the magnetic wall β^{1} + β^{2} and the symmetry wall β^{3} − β^{2} (D = 7), and
the electric wall β^{1} + β^{2} + β^{3} and the symmetry wall β^{4} − β^{3} (D ≥ 8),
is easily verified to be an integer submultiple of π only for D = 11, for which it is equal to π/3. From the point of view of the regularity of the billiard, the spacetime dimension D = 11 is thus privileged. This is of course also the dimension privileged by supersymmetry. It is quite intriguing that considerations a priori quite different (billiard regularity on the one hand, supersymmetry on the other hand) lead to the same conclusion that the gravity3form system is quite special in D = 11 spacetime dimensions.
For completeness, we here present the wall system relevant for the special case of D = 11. We obtain ten dominant wall forms, which we rename α_{1}, ⋯, α_{10},
then, defining a new collective index i = (m, 10), we see that the scalar products between these wall forms can be organized into the matrix
which can be identified with the Cartan matrix of the hyperbolic KacMoody algebra E_{10} that we have encountered in Section 4.10.2. We again display the corresponding Dynkin diagram in Figure 25, where we point out the explicit relation between the simple roots and the walls of the Einstein3form theory. It is clear that the nine dominant symmetry wall forms correspond to the simple roots α_{m} of the subalgebra \({\mathfrak {sl}(10,\,\mathbb R)}\). The enlargement to E_{10} is due to the tenth exceptional root realized here through the dominant electric wall form e_{123}.
5.2 Dynamics in the Cartan subalgebra
We have just learned that, in some cases, the group of reflections that describe the (possibly chaotic) dynamics in the BKLlimit is a Lorentzian Coxeter group \({\mathfrak C}\). In this section we fully exploit this algebraic fact and show that whenever \({\mathfrak C}\) is crystallographic, the dynamics takes place in the Cartan subalgebra \({\mathfrak h}\) of the Lorentzian KacMoody algebra \({\mathfrak g}\), for which \({\mathfrak C}\) is the Weyl group. Moreover, we show that the “billiard table” can be identified with the fundamental Weyl chamber in \({\mathfrak h}\).
5.2.1 Billiard dynamics in the Cartan subalgebra
5.2.1.1 Scale factor space and the wall system
Let us first briefly review some of the salient features encountered so far in the analysis. In the following we denote by \({{\mathcal M}_\beta}\) the Lorentzian “scale factor”space (or βspace) in which the billiard dynamics takes place. Recall that the metric in \({{\mathcal M}_\beta}\), induced by the EinsteinHilbert action, is a flat Lorentzian metric, whose explicit form in terms of the (logarithmic) scale factors reads
where d counts the number of physical spatial dimensions (see Section 2.5). The role of all other “offdiagonal” variables in the theory is to interrupt the freeflight motion of the particle, by adding walls in \({{\mathcal M}_\beta}\) that confine the motion to a limited region of scale factor space, namely a convex cone bounded by timelike hyperplanes. When projected onto the unit hyperboloid, this region defines a simplex in hyperbolic space which we refer to as the “billiard table”.
One has, in fact, more than just the walls. The theory provides these walls with a specific normalization through the Lagrangian, which is crucial for the connection to KacMoody algebras. Let us therefore discuss in somewhat more detail the geometric properties of the wall system. The metric, Equation (5.12), in scale factor space can be seen as an extension of a flat Euclidean metric in Cartesian coordinates, and reflects the Lorentzian nature of the vector space \({{\mathcal M}_\beta}\). In this space we may identify a pair of coordinates (β^{i}, φ) with the components of a vector \(\beta \in {{\mathcal M}_\beta}\), with respect to a basis {ū_{μ}} of \({{\mathcal M}_\beta}\), such that
The walls themselves are then defined by hyperplanes in this linear space, i.e., as linear forms ω = ω_{μ}σ^{μ}, for which ω = 0, where {σ^{μ}} is the basis dual to {ū^{μ}}. The pairing ω(β) between a vector \(\beta \in {{\mathcal M}_\beta}\) and a form \(\omega \in {\mathcal M}_\beta ^{\star}\) is sometimes also denoted by 〈ω, β〉, and for the two dual bases we have, of course,
We therefore find that the walls can be written as linear forms in the scale factors:
We call ω(β) wall forms. With this interpretation they belong to the dual space \({\mathcal M}_\beta ^{\star}\), i.e.,
From Equation (5.16) we may conclude that the walls bounding the billiard are the hyperplanes ω = 0 through the origin in \({{\mathcal M}_\beta}\) which are orthogonal to the vector with components ω^{μ} = G^{μv}ω_{v}.
It is important to note that it is the wall forms that the theory provides, as arguments of the exponentials in the potential, and not just the hyperplanes on which these forms ω vanish. The scalar products between the wall forms are computed using the metric in the dual space \({\mathcal M}_\beta ^{\star}\), whose explicit form was given in Section 2.5,
5.2.1.2 Scale factor space and the Cartan subalgebra
The crucial additional observation is that (for the “interesting” theories) the matrix A associated with the relevant walls ω_{A},
is a Cartan matrix, i.e., besides having 2’s on its diagonal, which is rather obvious, it has as offdiagonal entries nonpositive integers (with the property A_{AB} ≠ 0 ⇒ A_{BA} ≠ 0). This Cartan matrix is of course symmetrizable since it derives from a scalar product.
For this reason, one can usefully identify the space of the scale factors with the Cartan subalgebra \(\mathfrak{h}\) of the KacMoody algebra \(\mathfrak{g}{(A)}\) defined by A. In that identification, the wall forms become the simple roots, which span the vector space \(\mathfrak{h}^{\star}={\rm span}\{{\alpha _1},\, \cdots \,,\,{\alpha _r}\}\) dual to the Cartan subalgebra. The rank r of the algebra is equal to the number of scale factors β^{μ}, including the dilaton(s) if any ((β^{μ}) ≡ (β^{i}, φ)). This number is also equal to the number of walls since we assume the billiard to be a simplex. So, both A and μ run from 1 to r. The metric in \({{\mathcal M}_\beta}\), Equation (5.12), can be identified with the invariant bilinear form of \(\mathfrak{g}\), restricted to the Cartan subalgebra \(\mathfrak{h} \subset \mathfrak{g}\). The scale factors β^{μ} of \(\mathcal{M}_{\beta}\) become then coordinates h^{μ} on the Cartan subalgebra \(\mathfrak{h} \subset \mathfrak{g}(A)\).
The Weyl group of a KacMoody algebra has been defined first in the space \(\mathfrak{h}^{\star}\) as the group of reflections in the walls orthogonal to the simple roots. Since the metric is non degenerate, one can equivalently define by duality the Weyl group in the Cartan algebra \(\mathfrak{h}\) itself (see Section 4.7). For each reflection r_{i} on \(\mathfrak{h}^{\star}\) we associate a dual reflection \(r_i^ \vee\) as follows,
which is the reflection relative to the hyperplane α_{i}(β) = 〈α_{i}, β〉 = 0. This expression can be rewritten (see Equation (4.59)),
or, in terms of the scale factor coordinates β^{μ},
This is precisely the billiard reflection Equation (2.45) found in Section 2.4.
Thus, we have the following correspondence:
As we have also seen, the KacMoody algebra \(\mathfrak{g}(A)\) is Lorentzian since the signature of the metric Equation (5.12) is Lorentzian. This fact will be crucial in the analysis of subsequent sections and is due to the presence of gravity, where conformal rescalings of the metric define timelike directions in scale factor space.
We thereby arrive at the following important result [45, 46, 48]:
The dynamics of (a restricted set of) theories coupled to gravity can in the BKLlimit be mapped to a billiard motion in the Cartan subalgebra \(\mathfrak{h}\) of a Lorentzian KacMoody algebra \(\mathfrak{g}\).
5.2.2 The fundamental Weyl chamber and the billiard table
Let \(\mathcal{B}_{\mathcal{M}_{\beta}}\) denote the region in scale factor space to which the billiard motion is confined,
where the index A runs over all relevant walls. On the algebraic side, the fundamental Weyl chamber in \(\mathfrak{h}\) is the closed convex (half) cone given by
We see that the conditions α_{A}(h) ≥ 0 defining \(\mathcal{W}_{\mathfrak{h}}\) are equivalent, upon examination of Equation (5.22), to the conditions ω_{A}(β) ≥ 0 defining the billiard table \({{\mathcal B}_{{{\mathcal M}_\beta}}}\).
We may therefore make the crucial identification
which means that the particle geodesic is confined to move within the fundamental Weyl chamber of \(\mathfrak{h}\). From the billiard analysis in Section 2 we know that the piecewise motion in scalefactor space is controlled by geometric reflections with respect to the walls ω_{A}(β) = 0. By comparing with the dominant wall forms and using the correspondence in Equation (5.22) we may further conclude that the geometric reflections of the coordinates β^{μ}(τ) are controlled by the Weyl group in the Cartan subalgebra of \(\mathfrak{g}(A)\).
5.2.3 Hyperbolicity implies chaos
We have learned that the BKL dynamics is chaotic if and only if the billiard table is of finite volume when projected onto the unit hyperboloid. From our discussion of hyperbolic Coxeter groups in Section 3.5, we see that this feature is equivalent to hyperbolicity of the corresponding KacMoody algebra. This leads to the crucial statement [45, 46, 48]:
If the billiard region of a gravitational system in the BKLlimit can be identified with the fundamental Weyl chamber of a hyperbolic KacMoody algebra, then the dynamics is chaotic.
As we have also discussed above, hyperbolicity can be rephrased in terms of the fundamental weights Λ_{i} defined as
Just as the fundamental Weyl chamber in \(\mathfrak{h}^{\star}\) can be expressed in terms of the fundamental weights (see Equation (3.40)), the fundamental Weyl chamber in \(\mathfrak{h}\) can be expressed in a similar fashion in terms of the fundamental coweights:
As we have seen (Sections 3.5 and 4.8), hyperbolicity holds if and only if none of the fundamental weights are spacelike,
for all \(i \in \{1,\, \cdots \,,\,{\rm{rank}}\,\mathfrak{g}\}\).
5.2.3.1 Example: Pure gravity in D = 3 + 1 and \(A_1^{+ +}\)
Let us return once more to the example of pure fourdimensional gravity, i.e., the original “BKL billiard”. We have already found in Section 3 that the three dominant wall forms give rise to the Cartan matrix of the hyperbolic KacMoody algebra \(A_1^{+ +}\) [46, 48]. Since the algebra is hyperbolic, this theory exhibits chaotic behavior. In this example, we verify that the Weyl chamber is indeed contained within the lightcone by computing explicitly the norms of the fundamental weights.
It is convenient to first write the simple roots in the βbasis as follows¿
Since the Cartan matrix of \(A_1^{+ +}\) is symmetric, the relations defining the fundamental weights are
By solving these equations for Λ_{i} we deduce that the fundamental weights are
where in the last step we have written the fundamental weights in the βbasis. The norms may now be computed with the metric in root space and become
We see that Λ_{1} and Λ_{2} are timelike and that Λ_{3} is lightlike. Thus, the Weyl chamber is indeed contained inside the lightcone, the algebra is hyperbolic and the billiard is of finite volume, in agreement with what we already found [46].
5.3 Understanding the emerging KacMoody algebra
We shall now relate the KacMoody algebra whose fundamental Weyl chamber emerges in the BKLlimit to the Uduality group that appears upon toroidal dimensional reduction to three spacetime dimensions. We shall do this first in the case when \(\mathfrak{u}_3\) is a split real form. By this we mean that the real algebra \(\mathfrak{u}_3\) possesses the same ChevalleySerre presentation as \(\mathfrak{u}_3^{\mathbb{C}}\), but with coefficients restricted to be real numbers. This restriction is mathematically consistent because the coefficients appearing in the ChevalleySerre presentation are all reals (in fact, integers).
The fact that the billiard structure is preserved under reduction turns out to be very useful for understanding the emergence of “overextended” algebras in the BKLlimit. By computing the billiard in three spacetime dimensions instead of in maximal dimension, the relation to Uduality groups becomes particularly transparent and the computation of the billiard follows a similar pattern for all cases. We will see that if \(\mathfrak{u}_3\) is the algebra representing the internal symmetry of the nongravitational degrees of freedom in three dimensions, then the billiard is controlled by the Weyl group of the overextended algebra \(\mathfrak{u}_3^{++}\). The analysis is somewhat more involved when \(\mathfrak{u}_3\) is nonsplit, and we postpone a discussion of this until Section 7.
5.3.1 Invariance under toroidal dimensional reduction
It was shown in [41] that the structure of the billiard for any given theory is completely unaffected by dimensional reduction on a torus. In this section we illustrate this by an explicit example rather than in full generality. We consider the case of reduction of elevendimensional supergravity on a circle.
The compactification ansatz in the conventions of [35, 41] is
where μ, v = 0, 2, ⋯, 10, i.e., the compactification is performed along the first spatial direction^{Footnote 16}. We will refer to the new lowerdimensional fields \(\hat \varphi\) and \({\hat {\mathcal A}_\mu}\) as the dilaton and the KaluzaKlein (KK) vector, respectively. Quite generally, hatted fields are lowdimensional fields. The tendimensional Lagrangian becomes
where \(\hat{\mathcal{F}}^{(2)}=d\hat{\mathcal{A}}^{(1)}\) and \({\hat F^{(4)}},\,{\hat F^{(3)}}\) are the field strengths in ten dimensions originating from the elevendimensional 3form field strength F^{(4)} = dA^{(3)}.
Examining the new form of the metric reveals that the role of the scale factor β^{1}, associated to the compactified dimension, is now instead played by the tendimensional dilaton, \(\hat \varphi\). Explicitly we have
The nine remaining elevendimensional scale factors, β^{2}, ⋯, β^{10}, may in turn be written in terms of the new scale factors, \({\hat \beta ^a}\), associated to the tendimensional metric, ĝ_{μv}, and the dilaton in the following way:
We are interested in finding the dominant wall forms in terms of the new scale factors \({\hat \beta _2},\, \cdots \,,\,{\hat \beta _{10}}\) and \(\hat \varphi\). It is clear that we will have eight tendimensional symmetry walls,
which correspond to the eight simple roots of \(\mathfrak{sl}(9,\,\mathbb{R})\). Using Equation (5.35) and Equation (5.36) one may also check that the symmetry wall β^{2} — β^{1}, that was associated with the compactified direction, gives rise to an electric wall of the KaluzaKlein vector,
The metric in the dual space gets modified in a natural way,
i.e., the dilaton contributes with a flat spatial direction. Using this metric it is clear that \(\hat e_2^{\hat {\mathcal A}}\) has nonvanishing scalar product only with the second symmetry wall \({\hat s_2} = {\hat \beta ^3}  {\hat \beta ^2},\,(\hat e_2^{\hat {\mathcal A}}\vert{\hat s_2}) =  1\), and it follows that the electric wall of the KaluzaKlein vector plays the role of the first simple root of \(\mathfrak{sl}(10,\,\mathbb{R}),\,{\hat \alpha _1} \equiv \hat e_2^{\hat {\mathcal A}}\). The final wall form that completes the set will correspond to the exceptional node labeled “10” in Figure 25 and is now given by one of the electric walls of the NSNS 2form Â^{(2)}, namely
It is then easy to verify that this wall form has nonvanishing scalar product only with the third simple root \({\hat \alpha _3} = {\hat s_3},\,(\hat e_{23}^{{{\hat A}^{(2)}}}\vert{\hat s_3}) =  1\), as desired.
We have thus shown that the E_{10} structure is sufficiently rigid to withstand compactification on a circle with the new simple roots explicitly given by
This result is in fact true also for the general case of compactification on tori, T^{n}. When reaching the limiting case of three dimensions, all the nongravity wall forms correspond to the electric and magnetic walls of the axionic scalars. We will discuss this case in detail below.
For nontoroidal reductions the above analysis is drastically modified [166, 165]. The topology of the internal manifold affects the dominant wall system, and hence the algebraic structure in the lowerdimensional theory is modified. In many cases, the billiard of the effective compactified theory is described by a (nonhyperbolic) regular Lorentzian subalgebra of the original hyperbolic KacMoody algebra [98].
The walls are also invariant under dualization of a pform into a (D − p − 2)form; this simply exchanges magnetic and electric walls.
5.3.2 Iwasawa decomposition for split real forms
We will now exploit the invariance of the billiard under dimensional reduction, by considering theories that — when compactified on a torus to three dimensions — exhibit “hidden” internal global symmetries \({{\mathcal U}_3}\). By this we mean that the threedimensional reduced theory is described, after dualization of all vectors to scalars, by the sum of the EinsteinHilbert Lagrangian coupled to the Lagrangian for the nonlinear sigma model \({{\mathcal U}_3}/{\mathcal K}({{\mathcal U}_3})\). Here, \({\mathcal K}({{\mathcal U}_3})\) is the maximal compact subgroup defining the “local symmetries”. In order to understand the connection between the Uduality group \({{\mathcal U}_3}\) and the KacMoody algebras appearing in the BKLlimit, we must first discuss some important features of the Lie algebra \(\mathfrak{u}_3\).
Let \(\mathfrak{u}_3\) be a split real form, meaning that it can be defined in terms of the ChevalleySerre presentation of the complexified Lie algebra \(\mathfrak{u}_3^{\mathbb{C}}\) by simply restricting all linear combinations of generators to the real numbers ℝ. Let \(\mathfrak{h}_3\) be the Cartan subalgebra of \(\mathfrak{u}_3\) appearing in the ChevalleySerre presentation, spanned by the generators \(\alpha _1^ \vee, \, \cdots \,,\,\alpha _n^ \vee\). It is maximally noncompact (see Section 6). An Iwasawa decomposition of \(\mathfrak{u}_3\) is a direct sum of vector spaces of the following form,
where \(\mathfrak{k}_3\) is the “maximal compact subalgebra” of \(\mathfrak{u}_3\), and \(\mathfrak{n}_3\) is the nilpotent subalgebra spanned by the positive root generators E_{α}, ∀α ∈ Δ_{+}.
The corresponding Iwasawa decomposition at the group level enables one to write uniquely any group element as a product of an element of the maximally compact subgroup times an element in the subgroup whose Lie algebra is \(\mathfrak{h}_3\) times an element in the subgroup whose Lie algebra is \(\mathfrak{n}_3\). An arbitrary element \({\mathcal V}(x)\) of the coset \({{\mathcal U}_3}/{\mathcal K}({{\mathcal U}_3})\) is defined as the set of equivalence classes of elements of the group modulo elements in the maximally compact subgroup. Using the Iwasawa decomposition, one can go to the “Borel gauge”, where the elements in the coset are obtained by exponentiating only generators belonging to the Borel subalgebra,
In that gauge we have
where φ and χ are (sets of) coordinates on the coset space \({{\mathcal U}_3}/{\mathcal K}({{\mathcal U}_3})\). A Lagrangian based on this coset will then take the generic form (see Section 9)
where x denotes coordinates in spacetime and the “ellipses” denote correction terms that are of no relevance for our present purposes. We refer to the fields {φ} collectively as dilatons and the fields {χ} as axions. There is one axion field χ^{(α)} for each positive root α ∈ Δ_{+} and one dilaton field φ^{(i)} for each Cartan generator \(\alpha _i^ \vee \in \mathfrak{h}_3\).
The Lagrangian (5.45) coupled to the pure threedimensional EinsteinHilbert term is the key to understanding the appearance of the Lorentzian Coxeter group \(\mathfrak{u}_3^{++}\) in the BKLlimit.
5.3.3 Starting at the bottom — Overextensions of finitedimensional Lie algebras
To make the point explicit, we will again limit our analysis to the example of elevendimensional supergravity. Our starting point is then the Lagrangian for this theory compactified on an 8torus, T^{8}, to D = 2 + 1 spacetime dimensions (after all form fields have been dualized into scalars),
The second two terms in this Lagrangian correspond to the coset model Ɛ_{8(8)}/(Spin(16)/ℤ_{2}), where Ɛ_{8(8)} denotes the group obtained by exponentiation of the split form E_{8(8)} of the complex Lie algebra E_{8} and Spin(16)/ℤ_{2} is the maximal compact subgroup of Ɛ_{8(8)} [33, 134, 35]. The 8 dilatons \(\hat \varphi\) and the 120 axions χ^{(q)} are coordinates on the coset space^{Footnote 17}. Furthermore, the \({\alpha _q}(\hat \varphi)\) are linear forms on the elements of the Cartan subalgebra \(h = {\hat \varphi ^i}\alpha _i^ \vee\) and they correspond to the positive roots of E_{8(8)} ^{Footnote 18}. As before, we do not write explicitly the corrections to the curvatures \(d\hat \chi\) that appear in the compactification process. The entire set of positive roots can be obtained by taking linear combinations of the seven simple roots of \(\mathfrak{sl}(8,\,\mathbb{R})\) (we omit the “hatted” notation on the roots since there is no longer any risk of confusion),
and the exceptional root
These correspond exactly to the root vectors \({\vec b_{i,\,i + 1}}\) and \({\vec a_{123}}\) as they appear in the analysis of [35], except for the additional factor of \({1 \over {\sqrt 2}}\) needed to compensate for the fact that the aforementioned reference has an additional factor of 2 in the Killing form. Hence, using the Euclidean metric δ_{ij} (i, j = 1, ⋯, 8) one may check that the roots defined above indeed reproduce the Cartan matrix of E_{8}.
Next, we want to determine the billiard structure for this Lagrangian. As was briefly mentioned before, in the reduction from eleven to three dimensions all the nongravity walls associated to the elevendimensional 3form A^{(3)} have been transformed, in the same spirit as for the example given above, into electric and magnetic walls of the axionic scalars \(\hat \chi\). Since the terms involving the electric fields \({\partial _t}{\hat \chi ^{(i)}}\) possess no spatial indices, the corresponding wall forms do not contain any of the remaining scale factors \({\hat \beta ^9},\,{\hat \beta ^{10}}\), and are simply linear forms on the dilatons only. In fact the dominant electric wall forms are just the simple roots of E_{8},
The magnetic wall forms naturally come with one factor of \(\hat \beta\) since the magnetic field strength \({\partial _i}\hat \chi\) carries one spatial index. The dominant magnetic wall form is then given by
where \(\theta (\hat \varphi)\) denotes the highest root of E_{8} which takes the following form in terms of the simple roots,
Since we are in three dimensions there is no curvature wall and hence the only wall associated to the EinsteinHilbert term is the symmetry wall
coming from the threedimensional metric ĝ_{μv} (μ, v = 0, 9, 10). We have thus found all the dominant wall forms in terms of the lowerdimensional variables.
The structure of the corresponding Lorentzian KacMoody algebra is now easy to establish in view of our discussion of overextensions in Section 4.9. The relevant walls listed above are the simple roots of the (untwisted) overextension \(E_8^{+ +}\). Indeed, the relevant electric roots are the simple roots of E_{8}, the magnetic root of Equation (5.50) provides the affine extension, while the gravitational root of Equation (5.52) is the overextended root.
What we have found here in the case of elevendimensional supergravity also holds for the other theories with Uduality algebra \(\mathfrak{u}_3\) in 3 dimensions when \(\mathfrak{u}_3\) is a split real form. The Coxeter group and the corresponding KacMoody algebra are given by the untwisted overextension \(\mathfrak{u}_3^{++}\). This overextension emerges as follows [41]:
The dominant electric wall forms \({\hat e^{\hat \chi}}(\hat \varphi)\) for the supergravity theory in question are in onetoone correspondence with the simple roots of the associated Uduality algebra \(\mathfrak{u}_3\).
Adding the dominant magnetic wall form \({\hat m^{\hat \chi}}(\hat \beta, \,\hat \varphi) = {\hat \beta ^9}  \theta (\hat \varphi)\) corresponds to an affine extension \(\mathfrak{u}_3^{+}\) of \(\mathfrak{u}_3\).
Finally, completing the set of dominant wall forms with the symmetry wall \({\hat s_9}(\hat \beta) = {\hat \beta ^{10}}  {\hat \beta ^9}\), which is the only gravitational wall form existing in three dimensions, is equivalent to an overextension \(\mathfrak{u}_3^{++}\) of \(\mathfrak{u}_3\).
Thus we see that the appearance of overextended algebras in the BKLanalysis of supergravity theories is a generic phenomenon closely linked to hidden symmetries.
5.4 Models associated with split real forms
In this section we give a complete list of all theories whose billiard description can be given in terms of a KacMoody algebra that is the untwisted overextension of a split real form of the associated Uduality algebra (see Table 15). These are precisely the maximally oxidized theories introduced in [22] and further examined in [37]. These theories are completely classified by their global symmetry groups \({{\mathcal U}_3}\) arising in three dimensions. For the stringrelated theories the group \({{\mathcal U}_3}\) is the (classical version of) the Uduality symmetry obtained by combining the S and Tdualities in three dimensions [142]. Thereof the notation \({{\mathcal U}_3}\) for the global symmetry group in three dimensions. We extend the classification to the nonsplit case in Section 7.
Let us also note here that, as shown in [55], the billiard analysis sheds light on the problem of oxidation, i.e., the problem of finding the maximum spacetime dimension in which a theory with a given duality group in three dimensions can be reformulated. More on this question can be found in [118, 119].
6 FiniteDimensional Real Lie Algebras
In this section we explain the basic theory of real forms of finitedimensional Lie algebras. This material is somewhat technical and may therefore be skipped at a first reading. The theory of real forms of Lie algebras is required for a complete understanding of Section 7, which deals with the general case of KacMoody billiards for nonsplit real forms. However, for the benefit of the reader who wishes to proceed directly to the physical applications, we present a brief summary of the main points in the beginning of Section 7.
Our intention with the following presentation is to provide an accessible reference on the subject, directed towards physicists. We therefore consider this section to be somewhat of an entity of its own, which can be read independently of the rest of the paper. Consequently, we introduce Lie algebras in a rather different manner compared to the presentation of KacMoody algebras in Section 4, emphasizing here more involved features of the general structure theory of real Lie algebras rather than relying entirely on the ChevalleySerre basis and its properties. In the subsequent section, the reader will then see these two approaches merged, and used simultaneously to describe the billiard structure of theories whose Uduality algebras in three dimensions are given by arbitrary real forms.
We have adopted a rather detailed and explicit presentation. We do not provide all proofs, however, referring the reader to [93, 129, 133, 94] for more information (including definitions of basic Lie algebra theory concepts).
There are two main approaches to the classification of real forms of finitedimensional Lie algebras. One focuses on the maximal compact Cartan subalgebra and leads to Vogan diagrams. The other focuses on the maximal noncompact Cartan subalgebra and leads to TitsSatake diagrams. It is this second approach that is of direct use in the billiard analysis. However, we have chosen to present here both approaches as they mutually enlighten each other.
6.1 Definitions
Lie algebras are usually, in a first step at least, considered as complex, i.e., as complex vector spaces, structured by an antisymmetric internal bilinear product, the Lie bracket, obeying the Jacobi identity. If {T_{α}} denotes a basis of such a complex Lie algebra \({\mathfrak g}\) of dimension n (over ℂ), we may also consider \({\mathfrak g}\) as a real vector space of double dimension 2 n (over ℝ), a basis being given by {T_{α}, iT_{α}}. Conversely, if \({{\mathfrak g}_0}\) is a real Lie algebra, by extending the field of scalars from ℝ to ℂ, we obtain the complexification of \({{\mathfrak g}_0}\), denoted by \({{\mathfrak g}^{\mathbb C}}\), defined as:
Note that \({({{\mathfrak g}^{\mathbb C}})^{\mathbb R}} = {{\mathfrak g}_0} \oplus i{{\mathfrak g}_0}\) and \({\dim _{\mathbb R}}{({{\mathfrak g}^{\mathbb C}})^{\mathbb R}} = 2\,{\dim _{\mathbb R}}({{\mathfrak g}_0})\). When a complex Lie algebra \({\mathfrak g}\), considered as a real algebra, has a decomposition
with \({{\mathfrak g}_0}\) being a real Lie algebra, we say that \({{\mathfrak g}_0}\) is a real form of the complex Lie algebra \({\mathfrak g}\). In other words, a real form of a complex algebra exists if and only if we may choose a basis of the complex algebra such that all the structure constants become real. Note that while \({\mathfrak g}^{\mathbb R}\) is a real space, multiplication by a complex number is well defined on it since \({{\mathfrak g}_0} \oplus i{{\mathfrak g}_0} = {{\mathfrak g}_0}{\otimes _{\mathbb R}}{\mathbb C}\). As we easily see from Equation (6.2),
where a, b ∈ ℝ and X_{0}, Y_{0} ∈ \({Y_0} \in {{\mathfrak g}_0}\).
The Killing form is defined by
The Killing forms on \({\mathfrak g}^{\mathbb R}\) and \({\mathfrak g}^{\mathbb C}\) or \({{\mathfrak g}_0}\) are related as follows. If we split an arbitrary generator Z of \({\mathfrak g}\) according to Equation (6.2) as Z = X_{0} + iY_{0}, we may write:
Indeed, if \({\rm{a}}{{\rm{d}}_{\mathfrak g}}Z\) Z is a complex n × n matrix, \({\rm{a}}{{\rm{d}}_{{{\mathfrak g}^{\mathbb R}}}}({X_0} + i{Y_0})\) is a real 2n × 2n matrix:
6.2 A preliminary example: \({\mathfrak {sl}}(2,\,{\mathbb C})\)
Before we proceed to develop the general theory of real forms, we shall in this section discuss in detail some properties of the real forms of \({A_1} = {\mathfrak {sl}}(2,\,{\mathbb C})\). This is a nice example, which exhibits many properties that turn out not to be specific just to the case at hand, but are, in fact, valid also in the general framework of semisimple Lie algebras. The main purpose of subsequent sections will then be to show how to extend properties that are immediate in the case of \({\mathfrak {sl}}(2,\,{\mathbb C})\), to general semisimple Lie algebras.
6.2.1 Real forms of \({\mathfrak {sl}}(2,\,{\mathbb C})\)
The complex Lie algebra \({\mathfrak {sl}}(2,\,{\mathbb C})\) can be represented as the space of complex linear combinations of the three matrices
which satisfy the well known commutation relations
A crucial property of these commutation relations is that the structure constants defined by the brackets are all real. Thus by restricting the scalars in the linear combinations from the complex to the real numbers, we still obtain closure for the Lie bracket on real combinations of h, e and f, defining thereby a real form of the complex Lie algebra \({\mathfrak {sl}}(2,\,{\mathbb C})\): the real Lie algebra \({\mathfrak {sl}}(2,\,{\mathbb C})\) ^{Footnote 19}. As we have indicated above, this real form of \({\mathfrak {sl}}(2,\,{\mathbb C})\) is called the “split real form”.
Another choice of \({\mathfrak {sl}}(2,\,{\mathbb C})\) generators that, similarly, leads to a real Lie algebra consists in taking i times the Pauli matrices σ^{x}, σ^{y}, σ^{z}, i.e.,
The real linear combinations of these matrices form the familiar \({\mathfrak {su}}(2)\) Lie algebra (a real Lie algebra, even if some of the matrices using to represent it are complex). This real Lie algebra is nonisomorphic (as a real algebra) to \({\mathfrak {sl}}(2,\,{\mathbb R})\) as there is no real change of basis that maps {h, e, f} on a basis with the \({\mathfrak {su}}(2)\) commutation relations. Of course, the two algebras are isomorphic over the complex numbers.
6.2.2 Cartan subalgebras
Let \({\mathfrak h}\) be a subalgebra of \({\mathfrak {sl}}(2,\,{\mathbb R})\). We say that \({\mathfrak h}\) is a Cartan subalgebra of \({\mathfrak {sl}}(2,\,{\mathbb R})\) if it is a Cartan subalgebra of \({\mathfrak {sl}}(2,\,{\mathbb C})\) when the real numbers are replaced by the complex numbers. Two Cartan subalgebras \({\mathfrak {h}_1}\) and \({\mathfrak {h}_2}\) of \({\mathfrak {sl}}(2,\,{\mathbb R})\) are said to be equivalent (as Cartan subalgebras of \({\mathfrak {sl}}(2,\,{\mathbb R})\) if there is an automorphism a of \({\mathfrak {sl}}(2,\,{\mathbb R})\) such that \(a({\mathfrak {h}_1}) = {\mathfrak {h}_2}\).
The subspace ℝh constitutes clearly a Cartan subalgebra of \({\mathfrak {sl}}(2,\,{\mathbb R})\). The adjoint action of h is diagonal in the basis {e, f, h} and can be represented by the matrix
Another Cartan subalgebra of \({\mathfrak {sl}}(2,\,{\mathbb R})\) is given by ℝ(e − f) ≡ ℝτ^{y}, whose adjoint action with respect to the same basis is represented by the matrix
Contrary to the matrix representing ad_{h}, in addition to 0 this matrix has two imaginary eigenvalues: ±2i. Thus, there can be no automorphism a of \({\mathfrak {sl}}(2,\,{\mathbb R})\) such that τ^{y} = λa(h), λ ∈ ℝ since ad_{a(h)} has the same eigenvalues as ad_{h}, implying that the eigenvalues of λ ad_{a(h)} are necessarily real (λ ∈ ℝ).
Consequently, even though they are equivalent over the complex numbers since there is an automorphism in SL(2, ℂ) that connects the complex Cartan subalgebras ℂ h and ℂ τ^{y}, we obtain
The real Cartan subalgebras generated by h and τ^{y} are nonisomorphic over the real numbers.
6.2.3 The Killing form
The Killing form of SL(2, ℝ) reads explicitly
in the basis {e, f, h}. The Cartan subalgebra ℝh is spacelike while the Cartan subalgebra ℝτ^{y} is timelike. This is another way to see that these are inequivalent since the automorphisms of \({\mathfrak {sl}}(2,\,{\mathbb R})\) preserve the Killing form. The group \({\rm{Aut}[\mathfrak {sl}}(2,\,{\mathbb R})]\) of automorphisms of \({\mathfrak {sl}}(2,\,{\mathbb R})\) is SO(2, 1), while the subgroup Int[\({\rm{Int}[\mathfrak {sl}}(2,\,{\mathbb R})] \subset {\rm{Aut}[\mathfrak {sl}}(2,\,{\mathbb R})]\) of inner automorphisms is the connected component SO(2, 1)^{+} of SO(2, 1). All spacelike directions are equivalent, as are all timelike directions, which shows that all the Cartan subalgebras of \({\mathfrak {sl}}(2,\,{\mathbb R})\) can be obtained by acting on these two inequivalent particular ones by \({\rm{Int}[\mathfrak {sl}}(2,\,{\mathbb R})]\), i.e., the adjoint action of the group SL(2, ℝ). The lightlike directions do not define Cartan subalgebras because the adjoint action of a lighlike vector is nondiagonalizable. In particular ℝe and ℝf are not Cartan subalgebras even though they are Abelian.
By exponentiation of the generators h and τ^{y}, we obtain two subgroups, denoted \({\mathcal A}\) and \({\mathcal K}\):
The subgroup defined by Equation (6.14) is noncompact, the one defined by Equation (6.15) is compact; consequently the generator h is also said to be noncompact while τ^{y} is called compact.
6.2.4 The compact real form \({\mathfrak {su}}(2)\)
The Killing metric on the group \({\mathfrak {su}}(2)\) is negative definite. In the basis {τ^{x}, τ^{y}, τ^{z}}, it reads
The corresponding group obtained by exponentiation is SU(2), which is isomorphic to the 3sphere and which is accordingly compact. All directions in \({\mathfrak {su}}(2)\) are equivalent and hence, all Cartan subalgebras are SU(2) conjugate to ℝτ^{y}. Any generator provides by exponentiation a group isomorphic to ℝ/ℤ and is thus compact.
Accordingly, while \({\mathfrak {sl}}(2,\,{\mathbb R})\) admits both compact and noncompact Cartan subalgebras, the Cartan subalgebras of \({\mathfrak {su}}(2)\) are all compact. The real algebra \({\mathfrak {su}}(2)\) is called the compact real form of \({\mathfrak {sl}}(2,\,{\mathbb C})\). One often denotes the real forms by their signature. Adopting Cartan’s notation A_{1} for \({\mathfrak {sl}}(2,\,{\mathbb C})\), one has \({\mathfrak {sl}}(2,\,{\mathbb R}) \equiv {A_{1(1)}}\) and \({\mathfrak {su}}(2) \equiv {A_{1(3)}}\). We shall verify before that there are no other real forms of \({\mathfrak {sl}}(2,\,{\mathbb C})\).
6.2.5 \({\mathfrak {su}}(2)\) and \({\mathfrak {sl}}(2,\,{\mathbb R})\) compared and contrasted — The Cartan involution
Within \({\mathfrak {sl}}(2,\,{\mathbb C})\), one may express the basis vectors of one of the real subalgebras \({\mathfrak {su}}(2)\) or \({\mathfrak {sl}}(2,\,{\mathbb R})\) in terms of those of the other. We obtain, using the notations t = (e − f) and x = (e + f):
Let us remark that, in terms of the generators of \({\mathfrak {su}}(2)\), the noncompact generators x and h of \({\mathfrak {sl}}(2,\,{\mathbb R})\) are purely imaginary but the compact one t is real.
More precisely, if τ denotes the conjugation^{Footnote 20} of \({\mathfrak {sl}}(2,\,{\mathbb C})\) that fixes {τ^{x}, τ^{y}, τ^{z}}, we obtain:
or, more generally,
Conversely, if we denote by a the conjugation of \({\mathfrak {sl}}(2,\,{\mathbb C})\) that fixes the previous \({\mathfrak {sl}}(2,\,{\mathbb R})\) Cartan subalgebra in \({\mathfrak {sl}}(2,\,{\mathbb C})\), we obtain the usual complex conjugation of the matrices:
The two conjugations τ and a of \({\mathfrak {sl}}(2,\,{\mathbb C})\) associated with the real subalgebras \({\mathfrak {su}}(2)\) and \({\mathfrak {sl}}(2,\,{\mathbb R})\) of \({\mathfrak {sl}}(2,\,{\mathbb C})\) commute with each other. Each of them, trivially, fixes pointwise the algebra defining it and globally the other algebra, where it constitutes an involutive automorphism (“involution”).
The Killing form is neither positive definite nor negative definite on \({\mathfrak {sl}}(2,\,{\mathbb R})\): The symmetric matrices have positive norm squared, while the antisymmetric ones have negative norm squared. Thus, by changing the relative sign of the contributions associated with symmetric and antisymmetric matrices, one can obtain a bilinear form which is definite. Explicitly, the involution θ of \({\mathfrak {sl}}(2,\,{\mathbb R})\) defined by θ(X) = −X^{t} has the feature that
is positive definite. An involution of a real Lie algebra with that property is called a “Cartan involution” (see Section 6.4.3 for the general definition).
The Cartan involution θ is just the restriction to \({\mathfrak {sl}}(2,\,{\mathbb R})\) of the conjugation τ associated with the compact real form \({\mathfrak {su}}(2)\) since for real matrices X^{†} = X^{t}. One says for that reason that the compact real form \({\mathfrak {su}}(2)\) and the noncompact real form \({\mathfrak {sl}}(2,\,{\mathbb R})\) are “aligned”.
Using the Cartan involution θ, one can split \({\mathfrak {sl}}(2,\,{\mathbb R})\) as the direct sum
where \({\mathfrak k}\) is the subspace of antisymmetric matrices corresponding to the eigenvalue +1 of the Cartan involution while \({\mathfrak p}\) is the subspace of symmetric matrices corresponding to the eigenvalue −1. These are also eigenspaces of τ and given explicitly by \({\mathfrak k}={\mathbb R}t\) and \({\mathfrak p} = {\mathbb R}x \oplus {\mathbb R}h\). One has
i.e., the real form \({\mathfrak {sl}}(2,\,{\mathbb R})\) is obtained from the compact form \({\mathfrak {su}}(2)\) by inserting an “i” in front of the generators in \({\mathfrak p}\).
6.2.6 Concluding remarks
Let us close these preliminaries with some remarks.
 1.
The conjugation τ allows to define a Hermitian form on \({\mathfrak {sl}}(2,\,{\mathbb C})\):
$$X \bullet Y =  {\rm{Tr}}(Y\tau (X)).$$(6.24)  2.
Any element of the group SL(2, ℝ) can be written as a product of elements belonging to the subgroups \({\mathcal K}\), \({\mathcal A}\) and \({\mathcal N} = {\rm{Exp}}[{\mathbb R}e]\) (Iwasawa decomposition),
$${\rm{Exp}}[\theta \,t]\;{\rm{Exp}}[a\,h]\;{\rm{Exp}}[n\,e] = \left({\begin{array}{*{20}c} {{e^a}\cos \theta} & {n\,{e^a}\cos \theta + {e^{ a}}\sigma n\theta}\\ { {e^a}\sigma n\theta} & {{e^{ a}}\cos \theta  n\,{e^a}\sigma n\theta} \end{array}} \right){.}$$(6.25)  3.
Any element of \({\mathfrak p}\) is conjugated via \({\mathcal K}\) to a multiple of h,
$$\rho (\cos \alpha h + \sin\alpha x) = \left({\begin{array}{*{20}c} {\cos {{\alpha} \over 2}} & {\sin{{\alpha} \over 2}}\\ { \sin{{\alpha} \over 2}} & {\cos {{\alpha} \over 2}} \end{array}} \right)\rho \,h\;\left({\begin{array}{*{20}c} {\cos {{\alpha} \over 2}} & { \sin{{\alpha} \over 2}}\\ {\sin{{\alpha} \over 2}} & {\cos {{\alpha} \over 2}} \end{array}} \right),$$(6.26)so, denoting by \({\mathfrak a}={\mathbb R}h\) the (maximal) noncompact Cartan subalgebra of \({\mathfrak {sl}}(2,\,{\mathbb R})\), we obtain
$${\mathfrak{p}} = {\rm{Ad}}({\mathcal K}){\mathfrak{a}}{.}$$(6.27)  4.
Any element of SL(2, ℝ) can be written as the product of an element of \({\mathcal K}\) and an element of Exp[\({\rm{EXP[{\mathfrak p}]}}\)]. Thus, as a consequence of the previous remark, we have \(SL(2,\,{\mathbb R}) = {\mathcal {KAK}}\) (Cartan)^{Footnote 21}.
 5.
When the Cartan subalgebra of \({\mathfrak {sl}}(2,\,{\mathbb R})\) is chosen to be ℝ h, the root vectors are e and f. We obtain the compact element t, generating a nonequivalent Cartan subalgebra by taking the combination
$$t = e + \theta (e).$$(6.28)Similarly, the normalized root vectors associated with t are (up to a complex phase) E_{±2i} = ½(h ∓ ix):
$${[}t,\;{E_{2i}}] = 2i\,{E_{2i}},\qquad {[}t,\;{E_{ 2i}}] =  2i\;{E_{ 2i}},\qquad {[}{E_{2i}},\;{E_{ 2i}}] = i\;t.$$(6.29)Note that both the real and imaginary components of E_{±2i} are noncompact. They allow to obtain the noncompact Cartan generators h, x by taking the combinations
$$\cos \alpha \; h + \sin\;\alpha \,x = {e^{i\alpha}}{E_{2i}} + {e^{ i\alpha}}{E_{ 2i}}.$$(6.30)
6.3 The compact and split real forms of a semisimple Lie algebra
We shall consider here only semisimple Lie algebras. Over the complex numbers, Cartan subalgebras are “unique”^{Footnote 22}. These subalgebras may be defined as maximal Abelian subalgebras \({\mathfrak h}\) such that the transformations in ad[\({\rm{ad[{\mathfrak h}]}}\)] are simultaneously diagonalizable (over ℂ). Diagonalizability is an essential ingredient in the definition. There might indeed exist Abelian subalgebras of dimension higher than the rank (= dimension of Cartan subalgebras), but these would involve nondiagonalizable elements and would not be Cartan subalgebras^{Footnote 23}.
We denote the set of nonzero roots as Δ. One may complete the Chevalley generators into a full basis, the socalled CartanWeyl basis, such that the following commutation relations hold:
where H_{α} is defined by duality thanks to the Killing form B(X, Y) = Tr(ad X ad Y), which is nonsingular on semisimple Lie algebras:
and the generators are normalized according to (see Equation (6.43))
The generators E_{α} associated with the roots α (where α need not be a simple root) may be chosen such that the structure constants N_{α,β} satisfy the relations
where the scalar product between roots is defined as
The nonnegative integers p and q are such that the string of all vectors β + n α belongs to Δ for −p ≤ n ≤ q; they also satisfy the equation p − q = 2(βα)/(αα). A standard result states that for semisimple Lie algebras
from which we notice that the roots are real when evaluated on an H_{β}generator,
An important consequence of this discussion is that in Equation (6.32), the structure constants of the commutations relations may all be chosen real. Thus, if we restrict ourselves to real scalars we obtain a real Lie algebra \({{\mathfrak s}_0}\), which is called the split real form because it contains the maximal number of noncompact generators. This real form of \({\mathfrak g}\) reads explicitly
The signature of the Killing form on \({{\mathfrak s}_0}\) (which is real) is easily computed. First, it is positive definite on the real linear span \({{\mathfrak h}_0}\) of the H_{α} generators. Indeed,
Second, the invariance of the Killing form fixes the normalization of the E_{α} generators to one,
since^{Footnote 24}
Moreover, one has B(g_{α}, g_{β}) = 0 if α + β ≠ 0. Indeed ad[g_{α}] ad[g_{β}] maps g_{μ} into g_{μ+α+β}, i.e., in matrix terms ad[g_{α}] ad[g_{β}] has zero elements on the diagonal when α + β ≠ 0. Hence, the vectors E_{α} + E_{−α} are spacelike and orthogonal to the vectors E_{α} − E_{−α}, which are timelike. This implies that the signature of the Killing form is
The split real form \({{\mathfrak s}_0}\) of \({\mathfrak g}\) is “unique”.
On the other hand, it is not difficult to check that the linear span
also defines a real Lie algebra. An important property of this real form is that the Killing form is negative definite on it. Its signature is
This is an immediate consequence of the previous discussion and of the way \({{\mathfrak c}_0}\) is constructed. Hence, this real Lie algebra is compact^{Footnote 25}. For this reason, \({{\mathfrak c}_0}\) is called the “compact real form” of \({\mathfrak g}\). It is also “unique”.
6.4 Classical decompositions
6.4.1 Real forms and conjugations
The compact and split real Lie algebras constitute the two ends of a string of real forms that can be inferred from a given complex Lie algebra. As announced, this section is devoted to the systematic classification of these various real forms.
If \({{\mathfrak g}_0}\) is a real form of \({\mathfrak g}\), it defines a conjugation on \({\mathfrak g}\). Indeed we may express any \(Z \in {\mathfrak g}\) as Z = X_{0} + iY_{0} with \({X_0} \in {{\mathfrak g}_0}\) and \(i{Y_0} \in i{{\mathfrak g}_0}\), and the conjugation of \({\mathfrak g}\) with respect to \({{\mathfrak g}_0}\) is given by
Using Equation (6.3), it is immediate to verify that this involutive map is antilinear: \(\overline {\lambda Z} = \bar \lambda \bar Z\), where \(\bar \lambda\) is the complex conjugate of the complex number λ.
Conversely, if σ is a conjugation on \({\mathfrak g}\), the set \({{\mathfrak g}_\sigma}\) of elements of \({\mathfrak g}\) fixed by a provides a real form of \({\mathfrak g}\). Then σ constitutes the conjugation of \({\mathfrak g}\) with respect to \({{\mathfrak g}_\sigma}\). Thus, on \({\mathfrak g}\), real forms and conjugations are in onetoone correspondence. The strategy used to classify and describe the real forms of a given complex simple algebra consists of obtaining all the nonequivalent possible conjugations it admits.
6.4.2 The compact real form aligned with a given real form
Let \({{\mathfrak g}_0}\) be a real form of the complex semisimple Lie algebra \({{\mathfrak g}^{\mathbb C}} = {{\mathfrak g}_0}{\otimes _{\mathbb R}}{\mathbb C}\). Consider a compact real form \({{\mathfrak c}_0}\) of \({{\mathfrak g}^{\mathbb C}}\) and the respective conjugations τ and σ associated with \({{\mathfrak c}_0}\) and \({{\mathfrak g}_0}\). It may or it may not be that τ and σ commute. When they do, τ leaves \({{\mathfrak g}_0}\) invariant,
and, similarly, σ leaves \({{\mathfrak c}_0}\) invariant,
In that case, one says that the real form \({{\mathfrak g}_0}\) and the compact real form \({{\mathfrak c}_0}\) are “aligned”.
Alignment is not automatic. For instance, one can always dealign a compact real form by applying an automorphism to it while keeping \({{\mathfrak g}_0}\) unchanged. However, there is a theorem that states that given a real form \({{\mathfrak g}_0}\) of the complex Lie algebra \({{\mathfrak g}^{\mathbb C}}\), there is always a compact real form \({{\mathfrak c}_0}\) associated with it [93, 129]. As this result is central to the classification of real forms, we provide a proof in Appendix B, where we also prove the uniqueness of the Cartan involution.
We shall from now on always consider the compact real form aligned with the real form under study.
6.4.3 Cartan involution and Cartan decomposition
A Cartan involution θ of a real Lie algebra \({{\mathfrak g}_0}\) is an involutive automorphism such that the symmetric, bilinear form B^{θ} defined by
is positive definite. If the algebra \({B^\theta}\) is compact, a Cartan involution is trivially given by the identity.
A Cartan involution θ of the real semisimple Lie algebra \({{\mathfrak g}_0}\) yields the direct sum decomposition (called Cartan decomposition)
where \({{\mathfrak k}_0}\) and \({{\mathfrak p}_0}\) are the θeigenspaces of eigenvalues +1 and −1. Explicitly, the decomposition of a Lie algebra element is given by
The eigenspaces obey the commutation relations
from which we deduce that \(B({{\mathfrak k}_0},{{\mathfrak k}_0}) = 0\) because the mappings \({\rm{ab}}[{{\mathfrak k}_0}]\,{\rm{ab}}[{{\mathfrak p}_0}]\) map \({{\mathfrak p}_0}\) on \({{\mathfrak k}_0}\) and \({{\mathfrak k}_0}\) on \({{\mathfrak p}_0}\). Moreover \(\theta [{{\mathfrak k}_0}] = + {{\mathfrak k}_0}\) and \(\theta [{{\mathfrak p}_0}] =  {{\mathfrak p}_0}\), and hence \({B^\theta}({{\mathfrak k}_0},{{\mathfrak p}_0}) = 0\). In addition, since B^{θ} is positive definite, the Killing form B is negative definite on \({{\mathfrak k}_0}\) (which is thus a compact subalgebra) but is positive definite on \({{\mathfrak p}_0}\) (which is not a subalgebra).
Define in \({{\mathfrak g}^{\mathbb C}}\) the algebra \({{\mathfrak c}_0}\) by
It is clear that \({{\mathfrak c}_0}\) is also a real form of \({{\mathfrak g}^{\mathbb C}}\) and is furthermore compact since the Killing form restricted to it is negative definite. The conjugation τ that fixes \({{\mathfrak c}_0}\) is such that \(\tau (X) = X(X \in {{\mathfrak k}_0}),\,\tau (iY) = iY(Y \in {{\mathfrak p}_0})\) and hence \(\tau (Y) =  Y\,(Y \in {{\mathfrak p}_0})\). It leaves \({{\mathfrak g}_0}\) invariant, which shows that \({{\mathfrak c}_0}\) is aligned with \({{\mathfrak g}_0}\). One has
Conversely, let \({{\mathfrak c}_0}\) be a compact real form aligned with \({{\mathfrak g}_0}\) and τ the corresponding conjugation. The restriction θ of τ to \({{\mathfrak g}_0}\) is a Cartan involution. Indeed, one can decompose \({{\mathfrak g}_0}\) as in Equation (6.49), with Equation (6.51) holding since θ is an involution of \({{\mathfrak g}_0}\). Furthermore, one has also Equation (6.53), which shows that \({{\mathfrak k}_0}\) is compact and that B_{θ} is positive definite.
This shows, in view of the result invoked above that an aligned compact real form always exists, that any real form possesses a Cartan involution and a Cartan decomposition. If there are two Cartan involutions, θ and θ′, defined on a real semisimple Lie algebra, one can show that they are conjugated by an internal automorphism [93, 129]. It follows that any real semisimple Lie algebra possesses a “unique” Cartan involution.
On the matrix algebra \({\rm{ad}}[{{\mathfrak g}_0}]\), the Cartan involution is nothing else than minus the transposition with respect to the scalar product B^{θ},
Indeed, remembering that the transpose of a linear operator with respect to B^{θ} is defined by B^{θ}(X, AY) = B^{θ}(A^{T} X, Y), one gets
Since B_{θ} is positive definite, this implies, in particular, that the operator ad Y, with \(Y \in {{\mathfrak k}_0}\), is diagonalizable over the real numbers since it is symmetric, ad Y = (ad Y)^{T}.
An important consequence of this [93, 129] is that any real semisimple Lie algebra can be realized as a real matrix Lie algebra, closed under transposition. One can also show [93, 129] that the Cartan decomposition of the Lie algebra of a semisimple group can be lifted to the group via a diffeomorphism between \({{\mathfrak k}_0} \times {{\mathfrak p}_0} \mapsto {\mathcal G} = {\mathcal K}\exp [{{\mathfrak p}_0}]\), where \({\mathcal K}\) is a closed subgroup with \({{\mathfrak k}_0}\) as Lie algebra. It is this subgroup that contains all the topology of \({\mathcal G}\).
6.4.4 Restricted roots
Let \({{\mathfrak g}_0}\) be a real semisimple Lie algebra. It admits a Cartan involution θ that allows to split it into eigenspaces \({{\mathfrak k}_0}\) of eigenvalue +1 and \({{\mathfrak p}_0}\) of eigenvalue −1. We may choose in \({{\mathfrak p}_0}\) a maximal Abelian subalgebra \({{\mathfrak a}_0}\) (because the dimension of \({{\mathfrak p}_0}\) is finite). The set \(\{{\rm{ad}}\,H\vert H \in {{\mathfrak a}_0}\}\) is a set of symmetric transformations that can be simultaneously diagonalized on ℝ. Accordingly we may decompose \({{\mathfrak g}_0}\) into a direct sum of eigenspaces labelled by elements of the dual space \(a_0^{\ast}\):
One, obviously nonvanishing, subspace is g_{0}, which contains \({{\mathfrak a}_0}\). The other nontrivial subspaces define the restricted root spaces of \({{\mathfrak g}_0}\) with respect to \({{\mathfrak a}_0}\), of the pair \(({{\mathfrak g}_0}, \, {{\mathfrak a}_0})\). The λ that label these subspaces g_{λ} are the restricted roots and their elements are called restricted root vectors. The set of all λ is called the restricted root system. By construction the different g_{λ} are mutually B^{θ}orthogonal. The Jacobi identity implies that [g_{λ}, g_{μ}] ⊂ g_{λ+μ}, while \({{\mathfrak a}_0} \subset {{\mathfrak p}_0}\) implies that θg_{λ} = g_{−λ}. Thus if λ is a restricted root, so is −λ.
Let \({\mathfrak m}\) be the centralizer of \({{\mathfrak a}_0}\) in \({{\mathfrak k}_0}\). The space g_{0} is given by
If \({{\mathfrak t}_0}\) is a maximal Abelian subalgebra of \({\mathfrak m}\), the subalgebra \({{\mathfrak h}_0} = {{\mathfrak a}_0} \oplus {{\mathfrak t}_0}\) is a Cartan subalgebra of the real algebra \({{\mathfrak g}_0}\) in the sense that its complexification \({{\mathfrak h}^{\mathbb C}}\) is a Cartan subalgebra of \({{\mathfrak g}^{\mathbb C}}\). Accordingly we may consider the set of nonzero roots Δ of \({{\mathfrak g}^{\mathbb C}}\) with respect to \({{\mathfrak h}^{\mathbb C}}\) and write
The restricted root space g_{λ} is given by
and similarly
Note that the multiplicities of the restricted roots λ might be nontrivial even though the roots α are nondegenerate, because distinct roots α might yield the same restricted root when restricted to \({{\mathfrak a}_0}\).
Let us denote by Σ the subset of nonzero restricted roots and by V_{Σ} the subspace of \(a_0^{\ast}\) that they span. One can show [93, 129] that Σ is a root system as defined in Section 4. This root system need not be reduced. As for all root systems, one can choose a way to split the roots into positive and negative ones. Let Σ^{+} be the set of positive roots and
As Σ^{+} is finite, \({\mathfrak n}\) is a nilpotent subalgebra of \({{\mathfrak g}_0}\) and \({{\mathfrak a}_0} \oplus {\mathfrak n}\) is a solvable subalgebra.
6.4.5 Iwasawa and \({\mathcal {KAK}}\) decompositions
The Iwasawa decomposition provides a global factorization of any semisimple Lie group in terms of closed subgroups. It can be viewed as the generalization of the GramSchmidt orthogonalization process.
At the level of the Lie algebra, the Iwasawa decomposition theorem states that
Indeed any element X of \({{\mathfrak g}_0}\) can be decomposed as
The first term X_{0} belongs to \(g = {{\mathfrak a}_0} \oplus {\mathfrak m} \subset {{\mathfrak {a}_0}} \oplus {{\mathfrak {k}_0}}\), while the second term belongs to \({{\mathfrak k}_0}\), the eigenspace subspace of θeigenvalue +1. The third term belongs to \({\mathfrak n}\) since θX_{−λ} ∈ g_{λ}. The sum is furthermore direct. This is because one has obviously \({{\mathfrak {k}_0}} \cap {{\mathfrak {a}_0}} = 0\) as well as \({{\mathfrak {a}_0}} \cap {\mathfrak n} = 0\). Moreover, \({{\mathfrak {k}_0}} \cap {\mathfrak n}\) also vanishes because \(\theta {\mathfrak n} \cap {\mathfrak n} = 0\) as a consequence of \(\theta {\mathfrak n} = {\oplus _{\lambda \in \Sigma}} + {g_{ \lambda}}\).
The Iwasawa decomposition of the Lie algebra differs from the Cartan decomposition and is tilted with respect to it, in the sense that \({\mathfrak n}\) is neither in \({{\mathfrak k}_0}\) nor in \({{\mathfrak p}_0}\). One of its virtues is that it can be elevated from the Lie algebra \({{\mathfrak g}_0}\) to the semisimple Lie group \({\mathcal G}\). Indeed, it can be shown [93, 129] that the map
is a global diffeomorphism. Here, the subgroups \({\mathcal K},\,{\mathcal A}\) and \({\mathcal N}\) have respective Lie algebras \({\mathfrak k_0},\,{\mathfrak a_0}\). This decomposition is “unique”.
There is another useful decomposition of \({\mathcal G}\) in terms of a product of subgroups. Any two generators of \({{\mathfrak p}_0}\) are conjugate via internal automorphisms of the compact subgroup \({\mathcal K}\). As a consequence writing an element \(g \in {\mathcal G}\) as a product \(g = k\,{\rm{Exp}}[{{\mathfrak p}_0}]\), we may write \({\mathcal G} = {\mathcal K}{\mathcal A}{\mathcal K}\), which constitutes the socalled \({\mathcal K}{\mathcal A}{\mathcal K}\) decomposition of the group (also sometimes called the Cartan decomposition of the group although it is not the exponention of the Cartan decomposition of the algebra). Here, however, the writing of an element of \({\mathcal G}\) as product of elements of \({\mathcal K}\) and \({\mathcal A}\) is, in general, not unique.
6.4.6 θstable Cartan subalgebras
As in the previous sections, \({{\mathfrak g}_0}\) is a real form of the complex semisimple algebra \({\mathfrak g}\), σ denotes the conjugation it defines, τ the conjugation that commutes with σ, \({{\mathfrak c}_0}\) the associated compact aligned real form of \({\mathfrak g}\) and θ the Cartan involution. It is also useful to introduce the involution of \({\mathfrak g}\) given by the product στ of the commuting conjugations. We denote it also by θ since it reduces to the Cartan involution when restricted to \({{\mathfrak g}_0}\). Contrary to the conjugations σ and τ, θ is linear over the complex numbers. Accordingly, if we complexify the Cartan decomposition \({{\mathfrak g}_0} = {{\mathfrak k}_0} \oplus {{\mathfrak p}_0}\), to
with \({\mathfrak k} = {{\mathfrak k}_0}{\oplus _{\mathbb R}}{\mathbb C} = {{\mathfrak k}_0} \oplus i{{\mathfrak k}_0}\) and \({\mathfrak p} = {{\mathfrak p}_0}{\oplus _{\mathbb R}}{\mathbb C} = {{\mathfrak p}_0} \oplus i{{\mathfrak p}_0}\), the involution θ fixes \({\mathfrak k}\) pointwise while θ(X)=−X \(X \in {\mathfrak p}\).
Let \({{\mathfrak h}_0}\) be a θstable Cartan subalgebra of \({{\mathfrak g}_0}\), i.e., a subalgebra such that (i) \(\theta ({{\mathfrak h}_0}) \subset {{\mathfrak h}_0}\), and (ii) \({\mathfrak h} \equiv {\mathfrak h}_0^{\mathbb C}\) is a Cartan subalgebra of the complex algebra \({\mathfrak g}\). One can decompose \({{\mathfrak h}_0}\) into compact and noncompact parts,
We have seen that for real Lie algebras, the Cartan subalgebras are not all conjugate to each other; in particular, even though the dimensions of the Cartan subalgebras are all equal to the rank of \({\mathfrak g}\), the dimensions of the compact and noncompact subalgebras depend on the choice of \({{\mathfrak h}_0}\). For example, for \({\mathfrak {sl}(2,\,\mathbb R)}\), one may take \({{\mathfrak h}_0} = {\mathbb R}t\), in which case \({{\mathfrak t}_0} = 0,\,{{\mathfrak a}_0} = {{\mathfrak h}_0}\). Or one may take \({{\mathfrak h}_0} = {\mathbb R}{\tau ^y}\), in which case \({{\mathfrak t}_0} = {{\mathfrak h}_0},\,{{\mathfrak a}_0} = 0\).
One says that the θstable Cartan subalgebra \({{\mathfrak h}_0}\) is maximally compact if the dimension of its compact part \({{\mathfrak t}_0}\) is as large as possible; and that it is maximally noncompact if the dimension of its noncompact part \({{\mathfrak a}_0}\) is as large as possible. The θstable Cartan subalgebra \({{\mathfrak h}_0} = {{\mathfrak t}_0} \oplus {{\mathfrak a}_0}\) used above to introduce restricted roots, where \({{\mathfrak a}_0}\) is a maximal Abelian subspace of \({{\mathfrak p}_0}\) and \({{\mathfrak t}_0}\) a maximal Abelian subspace of its centralizer \({\mathfrak m}\), is maximally noncompact. If \({\mathfrak m} = 0\), the Lie algebra \({{\mathfrak g}_0}\) constitutes a split real form of \({{\mathfrak g}^{\mathcal C}}\). The real rank of \({{\mathfrak g}_0}\) is the dimension of its maximally noncompact Cartan subalgebras (which can be shown to be conjugate, as are the maximally compact ones [129]).
6.4.7 Real roots — Compact and noncompact imaginary roots
Consider a general θstable Cartan subalgebra \({{\mathfrak h}_0} = {{\mathfrak t}_0} \oplus {{\mathfrak a}_0}\), which need not be maximally compact or maximally non compact. A consequence of Equation (6.54) is that the matrices of the real linear transformations ad H are real symmetric for \(H \in {{\mathfrak a}_0}\) and real antisymmetric for \(H \in {{\mathfrak t}_0}\). Accordingly, the eigenvalues of ad H are real (and ad H can be diagonalized over the real numbers) when \(H \in {{\mathfrak a}_0}\), while the eigenvalues of ad H are imaginary (and ad H cannot be diagonalized over the real numbers although it can be diagonalized over the complex numbers) when \(H \in {{\mathfrak t}_0}\).
Let α be a root of \({\mathfrak g}\), i.e., a nonzero eigenvalue of ad \({\mathfrak h}\) where \({\mathfrak h}\) is the complexification of the θstable Cartan subalgebra \({{\mathfrak h}_0}\). As the values of the roots acting on a given H are the eigenvalues of ad H, we find that the roots are real on \({{\mathfrak a}_0}\) and imaginary on \({{\mathfrak t}_0}\). One says that a root is real if it takes real values on \({{\mathfrak h}_0} = {{\mathfrak t}_0} \oplus {{\mathfrak a}_0}\), i.e., if it vanishes on \({{\mathfrak t}_0}\). It is imaginary if it takes imaginary values on \({{\mathfrak h}_0}\), i.e., if it vanishes on \({{\mathfrak n}_0}\), and complex otherwise. These notions of “real” and “imaginary” roots should not be confused with the concepts with similar terminology introduced in Section 4 in the context of nonfinitedimensional KacMoody algebras.
If \({{\mathfrak h}_0}\) is a θstable Cartan subalgebra, its complexification \({\mathfrak h} = {{\mathfrak h}_0} {\oplus_{\mathbb R}}{\mathbb C} {{\mathfrak h}_0} \oplus i{{\mathfrak h}_0}\) is stable under the involutive authormorphism θ = τσ. One can extend the action of θ from \({\mathfrak h}\) to \({{\mathfrak h}^{\ast}}\) by duality. Denoting this transformation by the same symbol θ, one has
or, since θ^{2} = 1,
Let E_{α} be a nonzero root vector associated with the root α and consider the vector θE_{α}. One has
i.e., θ(g_{α}) = g_{θ(α)} because the roots are nondegenerate, i.e., all root spaces are onedimensional.
Consider now an imaginary root α. Then for all \(h \in {{\mathfrak h}_0}\) and \(a \in {{\mathfrak a}_0}\) we have α(h + a) = α(h), while θ(α) (h + a) = α(θ(h + a)) = α(h − a) = α(h); accordingly α = θ(α). Moreover, as the roots are nondegenerate, one has θE_{α} = ±E_{α}. Writing E_{α} as
it is easy to check that θE_{α} = +E_{α} implies that X_{α} and Y_{α} belong to \({{\mathfrak k}_0}\), while both are in \({{\mathfrak p}_0}\) if θE_{α} = −E_{α}. Accordingly, g_{α} is completely contained either in \({\mathfrak k} = {{\mathfrak k}_0} \oplus i{{\mathfrak k}_0}\) or in \({\mathfrak p} = {{\mathfrak p}_0} \oplus i{{\mathfrak p}_0}\). If \({g_\alpha} \subset {\mathfrak k}\), the imaginary root is said to be compact, and if \({g_\alpha} \subset {\mathfrak p}\) it is said to be noncompact.
6.4.8 Jumps between Cartan subalgebras — Cayley transformations
Suppose that β is an imaginary noncompact root. Consider a βroot vector \({E_\beta} \in {g_\beta} \subset {\mathfrak p}\). If this root is expressed according to Equation (6.70), then its conjugate, with respect to (the conjugation σ defined by) \({{\mathfrak g}_0}\), is
It belongs to g_{−β} because (using \(\forall H \in {{\mathfrak h}_0}:\sigma H = H\))
Hereafter, we shall denote σE_{β} by \({{\bar E}_\beta}\). The commutator
belongs to i \(i{\mathfrak k}_0\) since \(\sigma ([{E_\beta},\,{{\bar E}_\beta}]) = [{E_\beta},\,{{\bar E}_\beta}] =  [{E_\beta},\,{{\bar E}_\beta}]\) and can be written, after a renormalization of the generators E_{β}, as
Indeed as \({E_\beta} \in {\mathfrak p}\), we have \({{\bar E}_\beta} \in {\mathfrak p}\) and thus \(\theta {{\bar E}_\beta} =  {{\bar E}_\beta}\). This implies
The three generators \(\{{H_\beta},\,{E_\beta},\,{{\bar E}_\beta}\}\) therefore define an \({\mathfrak {sl}(2,\,\mathbb C)}\) subalgebra:
We may change the basis and take
whose elements belong to \({{\mathfrak g}_0}\) (since they are fixed by σ) and satisfy the commutation relations (6.8)
The subspace
constitutes a new real Cartan subalgebra whose intersection with \({{\mathfrak p}_0}\) has one more dimension.
Conversely, if β is a real root then θ(β) = −β. Let E_{β} be a root vector. Then \({{\bar E}_\beta}\) is also in g_{β} and hence proportional to E_{β}. By adjusting the phase of E_{β}, we may assume that E_{β} belongs to \({{\mathfrak g}_0}\). At the same time, θE_{β}, also in \({{\mathfrak g}_0}\), is an element of g_{−β}. Evidently, B(E_{β}, θE_{β}) = −B^{θ}(E_{β}, E_{β}) is negative. Introducing H_{β′} = 2/(β∣β)H_{β} (which is in \({{\mathfrak p}_0}\)), we obtain the \({\mathfrak {sl}(2,\,\mathbb R)}\) commutation relations
Defining the compact generator E_{β} + θE_{β}, which obviously belongs to \({{\mathfrak g}_0}\), we may build a new Cartan subalgebra of \({{\mathfrak g}_0}\):
whose noncompact subspace is now one dimension less than previously.
These two kinds of transformations — called Cayley transformations — allow, starting from a θstable Cartan subalgebra, to transform it into new ones with an increasing number of noncompact dimensions, as long as noncompact imaginary roots remain; or with an increasing number of compact dimensions, as long as real roots remain. Exploring the algebra in this way, we obtain all the Cartan subalgebras up to conjugacy. One can prove that the endpoints are maximally noncompact and maximally compact, respectively.
Theorem: Let \({{\mathfrak h}_0}\) be a θ stable Cartan subalgebra of \({{\mathfrak g}_0}\). Then there are no noncompact imaginary roots if and only if \({{\mathfrak h}_0}\) is maximally noncompact, and there are no real roots if and only if \({{\mathfrak h}_0}\) is maximally compact [129].
For a proof of this, note that we have already proven that if there are imaginary noncompact (respectively, real) roots, then \({{\mathfrak h}_0}\) is not maximally noncompact (respectively, compact). The converse is demonstrated in [129].
6.5 Vogan diagrams
Let \({{\mathfrak g}_0}\) be a real semisimple Lie algebra, \({\mathfrak g}\) its complexification, θ a Cartan involution leading to the Cartan decomposition
and \({{\mathfrak h}_0}\) a Cartan θstable subalgebra of \({{\mathfrak g}_0}\). Using, if necessary, successive Cayley transformations, we may build a maximally compact θstable Cartan subalgebra \({{\mathfrak h}_0} = {{\mathfrak t}_0} \oplus {{\mathfrak a}_0}\), with complexification \({\mathfrak h} = {\mathfrak t} \oplus {\mathfrak a}\). As usual we denote by Δ the set of (nonzero) roots of \({\mathfrak g}\) with respect to \({\mathfrak h}\). This set does not contain any real root, the compact dimension being assumed to be maximal.
From Δ we may define a positive subset Δ^{+} by choosing the first set of indices from a basis of \(i{{\mathfrak t}_0}\), and then the next set from a basis of \({{\mathfrak a}_0}\). Since there are no real roots, the roots in Δ^{+} have at least one nonvanishing component along \(i{{\mathfrak t}_0}\), and the first nonzero one of these components is strictly positive. Since θ = +1 on \({{\mathfrak t}_0}\), and since there are no real roots: θΔ^{+} = Δ^{+}. Thus θ permutes the simple roots, fixes the imaginary roots and exchanges in 2tuples the complex roots: it constitutes an involutive automorphism of the Dynkin diagram of \({\mathfrak g}\).
A Vogan diagram is associated to the triple \(({{\mathfrak g}_0},\,{{\mathfrak h}_{0,\,}}{\Delta ^ +})\) as follows. It corresponds to the standard Dynkin diagram of Δ^{+}, with additional information: the 2element orbits under θ are exhibited by joining the correponding simple roots by a double arrow and the 1element orbit is painted in black (respectively, not painted), if the corresponding imaginary simple root is noncompact (respectively, compact).
6.5.1 Illustration — The \({\mathfrak {sl}}(5,\,{\mathbb C})\) case
The complex Lie algebra \({\mathfrak {sl}}(5,\,{\mathbb C})\) can be represented as the algebra of traceless 5 × 5 complex matrices, the Lie bracket being the usual commutator. It has dimension 24. In principle, in order to compute the Killing form, one needs to handle the 24 × 24 matrices of the adjoint representation. Fortunately, the uniqueness (up to an overall factor) of the biinvariant quadratic form on a simple Lie algebra leads to the useful relation
The coefficient 10 appearing in this relation is known as the Coxeter index of \({\mathfrak {sl}}(5,\,{\mathbb C})\).
A CartanWeyl basis is obtained by taking the 20 nilpotent generators \({K^p}_q\) (with p ≠ q) corresponding to matrices, all elements of which are zero except the one located at the intersection of row p and column q, which is equal to 1,
and the four diagonal ones,
which constitute a Cartan subalgebra \({\mathfrak h}\).
The root space is easily described by introducing the five linear forms ϵ_{p}, acting on diagonal matrices d = diag(d_{1}, …, d_{5}) as follows:
In terms of these, the dual space \({{\mathfrak h}^{\ast}}\) of the Cartan subalgebra may be identified with the subspace
The 20 matrices \({K^p}_q\) are root vectors,
i.e., \({K^p}_q\) is a root vector associated to the root ϵ_{p} − ϵ_{q}.
6.5.1.1 \({\mathfrak {sl}}(5,\,{\mathbb R})\) and \({\mathfrak {su}}(5)\)
By restricting ourselves to real combinations of these generators we obtain the real Lie algebra \({\mathfrak {sl}}(5,\,{\mathbb R})\). The conjugation η that it defines on \({\mathfrak {sl}}(5,\,{\mathbb C})\) is just the usual complex conjugation. This \({\mathfrak {sl}}(5,\,{\mathbb R})\) constitutes the split real form \({{\mathfrak s}_0}\) of \({\mathfrak {sl}}(5,\,{\mathbb C})\). Applying the construction given in Equation (6.45) to the generators of \({\mathfrak {sl}}(5,\,{\mathbb R})\), we obtain the set of antihermitian matrices
defining a basis of the real subalgebra \({\mathfrak {su}}(5)\). This is the compact real form \(i{{\mathfrak c}_0}\) of \({\mathfrak {sl}}(5,\,{\mathbb C})\). The conjugation associated to this algebra (denoted by τ) is minus the Hermitian conjugation,
Since [η, τ] =0, τ induces a Cartan involution θ on \({\mathfrak {sl}}(5,\,{\mathbb R})\), providing a Euclidean form on the previous \({\mathfrak {sl}}(5,\,{\mathbb R})\) subalgebra
which can be extended to a Hermitian form on \({\mathfrak {sl}}(5,\,{\mathbb C})\),
Note that the generators i H_{k} and \(i({k^p}_q + {k^q}_p)\) are real generators (although described by complex matrices) since, e.g., \({(i{H_k})^\dagger} =  iH_k^\dagger\), i.e., \(\tau (i{H_k}) = i{H_k}\).
6.5.1.2 The other real forms
The real forms of \({\mathfrak {sl}}(5,\,{\mathbb C})\) that are not isomorphic to \({\mathfrak {sl}}(5,\,{\mathbb R})\) or \({\mathfrak {su}}(5)\) are isomorphic either to \({\mathfrak {su}}(3,\,2)\) or \({\mathfrak {su}}(4,\,1)\). In terms of matrices these algebras can be represented as
We shall call these ways of describing \({\mathfrak {su}}(p,\,q)\) the “natural” descriptions of \({\mathfrak {su}}(p,\,q)\). Introducing the diagonal matrix
the conjugations defined by these subalgebras are given by:
6.5.1.3 Vogan diagrams
The Dynkin diagram of \({\mathfrak {sl}}(5,\,{\mathbb C})\) is of A_{4} type (see Figure 26).
Let us first consider an \({\mathfrak {su}}(3,\,2)\) subalgebra. Diagonal matrices define a Cartan subalgebra whose all elements are compact. Accordingly all associated roots are imaginary. If we define the positive roots using the natural ordering ϵ_{1} > ϵ_{2} > ϵ_{3} > ϵ_{4} > ϵ_{5}, the simple roots α_{1} = ϵ_{1} − ϵ_{2}, α_{2} = ϵ_{2} − ϵ_{3}, α_{4} = ϵ_{4} − ϵ_{5} are compact but α_{3} = ϵ_{3} − ϵ_{4} is noncompact. The corresponding Vogan diagram is displayed in Figure 27.
However, if instead of the natural order we define positive roots by the rule ϵ_{1} > ϵ_{2} > ϵ_{4} > ϵ_{5} > ϵ_{3}, the simple positive roots are \({\tilde \alpha _1} = {\epsilon _1}  {\epsilon _2}\) and \({\tilde \alpha _3} = {\epsilon _4}  {\epsilon _5}\) which are compact, and \({\tilde \alpha _2} = {\epsilon _2}  {\epsilon _4}\) and \({\tilde \alpha _4} = {\epsilon _5}  {\epsilon _3}\) which are noncompact. The associated Vogan diagram is shown in Figure 28.
Alternatively, the choice of order ϵ_{1} > ϵ_{5} > ϵ_{3} > ϵ_{4} > ϵ_{2} leads to the diagram in Figure 29.
There remain seven other possibilities, all describing the same subalgebra \({\mathfrak {su}}(3,\,2)\). These are displayed in Figure 30.
In a similar way, we obtain four different Vogan diagrams for \({\mathfrak {su}}(4,\,1)\), displayed in Figure 31.
Finally we have two nonisomorphic Vogan diagrams associated with \({\mathfrak {su}}(5)\) and \({\mathfrak {sl}}(5,\,{\mathbb R})\). These are shown in Figure 32.
6.5.2 The Borel and de Siebenthal theorem
As we just saw, the same real Lie algebra may yield different Vogan diagrams only by changing the definition of positive roots. But fortunately, a theorem of Borel and de Siebenthal tells us that we may always choose the simple roots such that at most one of them is noncompact [129]. In other words, we may always assume that a Vogan diagram possesses at most one black dot.
Furthermore, assume that the automorphism associated with the Vogan diagram is the identity (no complex roots). Let {α_{p}} be the basis of simple roots and {Λ_{q}} its dual basis, i.e., (Λ_{q}∣α_{p}) = δ_{p q}. Then the single painted simple root α_{p} may be chosen so that there is no q with (Λ_{p} − Λ_{q} ∣Λ_{q}) > 0. This remark, which limits the possible simple root that can be painted, is particularly helpful when analyzing the real forms of the exceptional groups. For instance, from the Dynkin diagram of E_{8} (see Figure 33), it is easy to compute the dual basis and the matrix of scalar products Bp q = (Λp − Λq ∣Λq).
We obtain
from which we see that there exist, besides the compact real form, only two other nonisomorphic real forms of E_{8}, described by the Vogan diagrams in Figure 34^{Footnote 26}.
6.5.3 Cayley transformations in su(3, 2)
Let us now illustrate the Cayley transformations. For this purpose, consider again \({\mathfrak {su}}(3,\,2)\) with the imaginary diagonal matrices as Cartan subalgebra and the natural ordering of the ϵ_{k} defining the positive roots. As we have seen, α_{3} = ϵ_{3} − ϵ_{4} is an imaginary noncompact root. The associated \({\mathfrak {sl}}(2,\,{\mathbb C})\) generators are
From the action of α_{3} on the Cartan subalgebra D = span{i H_{k}, k = 1, …, 4}, we may check that
and that \({H\prime} = ({E_{{\alpha _3}}} + \overline {{E_{{\alpha _3}}}}) = (K_4^3 + K_3^4)\) is such that θH′ = −H′ and σH′ = H′. Moreover H′ commutes with ker(α_{3}∣D). Thus
constitutes a θstable Cartan subalgebra with one noncompact dimension H′. Indeed, we have B(H′, H′) = 20. If we compute the roots with respect to this new Cartan subalgebra, we obtain twelve complex roots (expressed in terms of their components in the basis dual to the one implicitly defined by Equations (6.97) and (6.98),
six imaginary roots
and a pair of real roots ±(0, 0, 0, 2).
Let us first consider the Cayley transformation obtained using, for instance, the real root (0, 0, 0, 2). An associated root vector, belonging to \({{\mathfrak g}_0}\), reads
The corresponding compact Cartan generator is
which, together with the three generators in Equation (6.97), provide a compact Cartan subalgebra of \({\mathfrak {su}}(3,\,2)\).
If we consider instead the imaginary roots, we find for instance that \(K_2^5 =  \tilde \theta K_2^5\) is a noncompact complex root vector corresponding to the root β = i(1, −2, −2, 0). It provides the noncompact generator \(K_5^2 = + \,K_2^5\) which, together with
generates a maximally noncompact Cartan subalgebra of \({\mathfrak {su}}(3,\,2)\). A similar construction can be done using, for instance, the roots ±i(1, 0, 2, 0), but not with the roots ±i(2, −2, 0, 0) as their corresponding root vectors \(K_2^1\) and \(K_1^2\) are fixed by \(\tilde \theta\) and thus are compact.
6.5.4 Reconstruction
We have seen that every real Lie algebra leads to a Vogan diagram. Conversely, every Vogan diagram defines a real Lie algebra. We shall sketch the reconstruction of the real Lie algebras from the Vogan diagrams here, referring the reader to [129] for more detailed information.
Given a Vogan diagram, the reconstruction of the associated real Lie algebra proceeds as follows. From the diagram, which is a Dynkin diagram with extra information, we may first construct the associated complex Lie algebra, select one of its Cartan subalgebras and build the corresponding root system. Then we may define a compact real subalgebra according to Equation (6.45).
We know the action of θ on the simple roots. This implies that the set Δ of all roots is invariant under θ. This is proven inductively on the level of the roots, starting from the simple roots (level 1). Suppose we have proven that the image under θ of all the positive roots, up to level n are in Δ. If γ is a root of level n +1, choose a simple root α such that (γ∣α) > 0. Then the Weyl transformed root s_{α}γ = β is also a positive root, but of smaller level. Since θ(α) and θ(β) are then known to be in Δ, and since the involution acts as an isometry, θ(γ) = s_{θ(α)}(θ(β)) is also in Δ.
One can transfer by duality the action of θ on \({{\mathfrak h}^{\ast}}\) to the Cartan subalgebra \({\mathfrak h}\), and then define its action on the root vectors associated to the simple roots according to the rules
These rules extend θ to an involution of \({\mathfrak g}\).
This involution is such that θE_{α} = a_{α}E_{θ[α]}, with a_{α} = ±1 ^{Footnote 27}. Furthermore it globally fixes \({{\mathfrak c}_0},\,\theta {{\mathfrak c}_0} = {{\mathfrak c}_0}\). Let \({\mathfrak k}\) and \({\mathfrak p}\) be the +1 or −1 eigenspaces of θ in \({\mathfrak g} = {\mathfrak k} \oplus {\mathfrak p}\). Define \({{\mathfrak k}_0} = {{\mathfrak c}_0} \cap {\mathfrak k}\) and \({{\mathfrak p}_0} = i({{\mathfrak c}_0} \cap {\mathfrak p})\) so that \({{\mathfrak c}_0} = {{\mathfrak k}_0} \oplus i\,{{\mathfrak p}_0}\). Set
Using \(\theta {{\mathfrak c}_0} = {{\mathfrak c}_0}\), one then verifies that \(i{{\mathfrak g}_0}\) constitutes the desired real form of \({\mathfrak g}\) [129].
6.5.5 Illustrations: \({\mathfrak {sl}}(4,\,{\mathbb R})\) versus \({\mathfrak {sl}}(2,\,{\mathbb H})\)
We shall exemplify the reconstruction of real algebras from Vogan diagrams by considering two examples of real forms of \({\mathfrak {sl}}(4,\,{\mathbb C})\). The diagrams are shown in Figure 35.
The θ involutions they describe are (the upper signs correspond to the lefthand side diagram, the lower signs to the righthand side diagram):
Using the commutations relations
we obtain
Let us consider the lefthand side diagram. The corresponding +1 θeigenspace \({\mathfrak k}\) has the following realisation,
and the −1 θeigenspace \({\mathfrak p}\) is given by
The intersection \({{\mathfrak c}_0} \cap {\mathfrak k}\) then leads to the \({\mathfrak {so}}(4,\,{\mathbb R}) = {\mathfrak {so}}(3,\,{\mathbb R}) \oplus {\mathfrak {so}}(3,\,{\mathbb R})\) algebra
and the remaining noncompact generator subspace \({{\mathfrak p}_0} = i({{\mathfrak c}_0} \cap {\mathfrak p})\) becomes
Doing the same exercise for the second diagram, we obtain the real algebra \({\mathfrak {sl}}(2,\,{\mathbb H})\) with \({{\mathfrak k}_0} = {\mathfrak {so}}(5,\,{\mathbb R}) = {\mathfrak {sp}}(4,\,{\mathbb R})\), which is a 10parameter compact subalgebra, and \({{\mathfrak p}_0}\) given by
6.5.6 A pictorial summary — All real simple Lie algebras (Vogan diagrams)
The following tables provide all real simple Lie algebras and the corresponding Vogan diagrams. The restrictions imposed on some of the Lie algebra parameters eliminate the consideration of isomorphic algebras. See [129] for the derivation.
Using these diagrams, the matrix I_{p, q} defined by Equation (6.93), and the three matrices
we may check that the involutive automorphisms of the classical Lie algebras are all conjugate to one of the types listed in Table 25.
For completeness we remind the reader of the definitions of matrix algebras (\({\mathfrak {su}}(p,\,q)\) has been defined in Equation (6.93)):
Alternative definitions are:
For small dimensions we have the following isomorphisms:
6.6 TitsSatake diagrams
The classification of real forms of a semisimple Lie algebra, using Vogan diagrams, rests on the construction of a maximally compact Cartan subalgebra. On the other hand, the Iwasawa decomposition emphasizes the role of a maximally noncompact Cartan subalgebra. The consideration of these Cartan subalgebras leads to another way to classify real forms of semisimple Lie algebras, developed mainly by Araki [5], and based on socalled TitsSatake diagrams [161, 155].
6.6.1 Example 1: \({\mathfrak {su}}(3,\,2)\)
6.6.1.1 Diagonal description
At the end of Section 6.5.3, we obtained a matrix representation of a maximally noncompact Cartan subalgebra of \({\mathfrak {su}}(3,\,2)\) in terms of the natural description of this algebra. To facilitate the forthcoming discussion, we find it useful to use an equivalent description, in which the matrices representing this Cartan subalgebra are diagonal, as this subalgebra will now play a central role. It is obtained by performing a similarity transformation X ↦ S^{T} X S, where
In this new “diagonal” description, the conjugation σ (see Equation (6.94)) becomes
where
The Cartan involution has the following realisation:
In terms of the four matrices introduced in Equation (6.84), the generators defining this Cartan subalgebra \({\mathfrak h}\) reads
Let us emphasize that we have numbered the basis generators of \({\mathfrak h} = {\mathfrak a} \oplus {\mathfrak t}\) by first choosing those in \({\mathfrak a}\), then those in \({\mathfrak t}\).
6.6.1.2 Cartan involution and roots
The standard matrix representation of \({\mathfrak {su}}(5)\) constitutes a compact real Lie subalgebra of \({\mathfrak {sl}}(5,\,{\mathbb C})\) aligned with the diagonal description of the real form \({\mathfrak {su}}(3,\, 2)\). Moreover, its Cartan subalgebra \({{\mathfrak h}_1}\) generated by purely imaginary combinations of the four diagonal matrices H_{k} is such that its complexification \({{\mathfrak h}^{\mathbb C}}\) contains \({\mathfrak h}\). Accordingly, the roots it defines act both on \({{\mathfrak h}_0}\) and \({\mathfrak h}\). Note that on \({\mathfrak h}_{\mathbb R} = i\, {\mathfrak h}_0\), the roots take only real values.
Our first task is to compute the action of the Cartan involution θ on the root lattice. With this aim in view, we introduce two distinct bases on \({\mathfrak h}_{\mathbb R}^{\ast}\). The first one is {F^{1}, F^{2}, F^{3}, F^{4}}, which is dual to the basis {H_{1}, H_{2}, H_{3}, H_{4}} and is adapted to the relation \({\mathfrak h}_{\mathbb R} = i\, {\mathfrak h}_0\) The second one is {f^{1}, f^{2}, f^{3}, f^{4}}, dual to the basis {h_{1}, h_{2}, −ih_{3}, −ih_{4}}, which is adapted to the decomposition \({{\mathfrak h}_{\mathbb R}} = {\mathfrak a} \oplus i{\mathfrak t}\). The Cartan involution acts on these root space bases as
From the relations (6.130) it is easy to obtain the expression of the {F^{k}} (k =1, ⋯, 4) in terms of the {f^{k}} and thus also the expressions for the simple roots α_{1} = 2F^{1} − F^{2}, α_{2} = −F^{1} + 2F^{2} − F^{3}, α_{3} = −F^{2} + 2F^{3} − F^{4} and α_{4} = −F^{3} + 2F^{4}, defined by \({\mathfrak {h}_0}\),
It is then straightforward to obtain the action of θ on the roots, which, when expressed in terms of the \({\mathfrak {h}_0}\) simple roots themselves, is given by
We see that the root α_{3} is real while α_{1}, α_{2} and α_{4} are complex As a check of these results, we may, for instance, verify that
In fact, this kind of computation provides a simpler way to obtain Equation (6.133)
The basis {f^{1}, f^{2}, f^{3}, f^{4}} allows to define a different ordering on the root lattice, merely by considering the corresponding lexicographic order. In terms of this new ordering we obtain for instance α_{1} < 0 since the first nonzero component of α_{1} (in this case −1 along f^{2}) is strictly negative. Similarly, one finds α_{2} < 0, α_{3} > 0, α_{4} < 0, α_{1} + α_{2} < 0, α_{2} + α_{3} > 0, α_{3} + α_{4} > 0, α_{1} + α_{2} + α_{3} > 0, α_{2} + α_{3} + α_{4} > 0, α_{1} + α_{2} + α_{3} + α_{4} > 0. A basis of simple roots, according to this ordering, is given by
(We have put \({{\tilde \alpha}_4}\) in fourth position, rather than in second, to follow usual conventions.) The action of θ on this basis reads
These new simple roots are now all complex.
6.6.1.3 Restricted roots
The restricted roots are obtained by considering only the action of the roots on the noncompact Cartan generators h_{1} and h_{2}. The twodimensional vector space spanned by the restricted roots can be identified with the subspace spanned by f_{1} and f_{2}; one simply projects out f_{3} and f_{4}. In the notations β_{1} = f_{1} − f_{2} and β_{2} = f_{2}, one gets as positive restricted roots:
which are the positive roots of the (BC)_{2} (nonreduced) root system. The first four roots are degenerate twice, while the last two roots are nondegenerate. For instance, the two simple roots \({{\tilde \alpha}_1}\) and \({{\tilde \alpha}_4}\) project on the same restricted root β_{1}, while the two simple roots \({{\tilde \alpha}_2}\) and \({{\tilde \alpha}_3}\) project on the same restricted root β_{2}.
Counting multiplicities, there are ten restricted roots — the same number as the number of positive roots of \({\mathfrak {sl}}(5,\,{\mathbb C})\). No root of \({\mathfrak {sl}}(5,\,{\mathbb C})\) projects onto zero. The centralizer of \({\mathfrak a}\) consists only of \({\mathfrak a} \oplus {\mathfrak t}\).
6.6.2 Example 2: \({\mathfrak {su}}(4,\, 1)\)
6.6.2.1 Diagonal description
Let us now perform the same analysis within the framework of \({\mathfrak {su}}(4,\, 1)\). Starting from the natural description (6.92) of \({\mathfrak {su}}(4,\, 1)\), we first make a similarity transformation using the matrix
so that a maximally noncompact Cartan subalgebra can be taken to be diagonal and is explicitly given by
The corresponding \({\mathfrak {su}}(4,\, 1)\) in the \({\mathfrak {sl}}(5,\,{\mathbb C})\) algebra is still aligned with the natural matrix representation of \({\mathfrak {su}}(5)\). The Cartan involution is given by X ↦ Ĩ_{4,1} X Ĩ_{4,1} where Ĩ_{4,1} = S^{T} I_{4,1} S. One has \({\mathfrak h}={\mathfrak a} \oplus {\mathfrak t}\) where the noncompact part \({\mathfrak a}\) is onedimensional and spanned by h_{1}, while the compact part t is threedimensional and spanned by h_{2}, h_{3} and h_{4}.
6.6.2.2 Cartan involution and roots
In terms of the f^{i}’s, the standard simple roots now read
The Cartan involution acts as
showing that α_{1} and α_{2} are imaginary, α_{4} is real, while α_{3} is complex.
A calculation similar to the one just described above, using as ordering rules the lexicographic ordering defined by the dual of the basis in Equation (6.139), leads to the new system of simple roots,
which transform as
under the Cartan involution. Note that in this system, the simple roots \({{\tilde \alpha}_2}\) and \({{\tilde \alpha}_3}\) are imaginary and hence fixed by the Cartan involution, while the other two simple roots are complex.
6.6.2.3 Restricted roots
The restricted roots are obtained by considering the action of the roots on the single noncompact Cartan generator h_{1}. The onedimensional vector space spanned by the restricted roots can be identified with the subspace spanned by f_{1}; one now simply projects out f_{2}, f_{3} and f_{4}. With the notation β_{1} = f_{1}, we get as positive restricted roots
which are the positive roots of the (BC)_{1} (nonreduced) root system. The first root is six times degenerate, while the second one is nondegenerate. The simple roots \({{\tilde \alpha}_1}\) and \({{\tilde \alpha}_4}\) project on the same restricted root β_{1}, while the imaginary root \({{\tilde \alpha}_2}\) and \({{\tilde \alpha}_3}\) project on zero (as does also the nonsimple, positive, imaginary root \({{\tilde \alpha}_2} + {{\tilde \alpha}_3}\)).
Let us finally emphasize that the centralizer of \({\mathfrak a}\) in \({\mathfrak {su}}(4,\, 1)\) is now given by \({\mathfrak a} \oplus {\mathfrak m}\), where \({\mathfrak m}\) is the center of \({\mathfrak a}\) in \({\mathfrak k}\) (i.e., the subspace generated by the compact generators that commute with H_{4}) and contains more than just the three compact Cartan generators h_{2}, h_{3} and h_{4}. In fact, m involves also the root vectors E_{β} whose roots restrict to zero. Explicitly, expressed in the basis of Equation (6.85), these roots read β = ϵ_{p} − ϵ_{q} with p, q = 1, 2, or 3 and are orthogonal to α_{4}. The algebra m constitutes a rank 3, 9dimensional Lie algebra, which can be identified with \({\mathfrak {su}}(3) \oplus {\mathfrak u}(1)\).
6.6.3 TitsSatake diagrams: Definition
We may associate with each of the constructions of these simple root bases a TitsSatake diagram as follows. We start with a Dynkin diagram of the complex Lie algebra and paint in black (●) the imaginary simple roots, i.e., the ones fixed by the Cartan involution. The others are represented by a white vertex (○). Moreover, some double arrows are introduced in the following way. It can be easily proven (see Section 6.6.4) that for real semisimple Lie algebras, there always exists a basis of simple roots B that can be split into two subsets: B_{0} = {α_{r+1}, …, α_{l}} whose elements are fixed by θ (they correspond to the black vertices) and B B_{0} = {α_{1}, …, α_{r}} (corresponding to white vertices) such that
where π is an involutive permutation of the r indices of the elements of B B_{0}. Accordingly, π contains cycles of one or two elements. In the TitsSatake diagram, we connect by a double arrow all pairs of distinct simple roots α_{k} and α_{π}_{(k)} in the same twocycle orbit. For instance, for \({\mathfrak {su}}(3,\, 2)\) and \({\mathfrak {su}}(4,\, 1)\), we obtain the diagrams in Figure 36.
6.6.4 Formal considerations
TitsSatake diagrams provide a lot of information about real semisimple Lie algebras. For instance, we can read from them the full action of the Cartan involution as we now briefly pass to show. More information may be found in [5, 93].
The Cartan involution allows one to define a closed subsystem^{Footnote 28} Δ_{0} of Δ:
which is the system of imaginary roots. These project to zero when restricted to the maximally noncompact Cartan subalgebra. As we have seen in the examples, it is useful to use an ordering adapted to the Cartan involution. This can be obtained by considering a basis of \({\mathfrak h}\) constituted firstly by elements of \({\mathfrak a}\) followed by elements of \({\mathfrak t}\). If we use the lexicographic order defined by the dual of this basis, we obtain a root ordering such that if α ∉ Δ_{0} is positive, θ[α] is negative since the real part comes first and changes sign. Let B be a simple root basis built with respect to this ordering and let B_{0} = B ∩ Δ_{0}. Then we have
The subset B_{0} is a basis for Δ_{0}. To see this, let B B_{0} = {α_{1}, …, α_{r}}. If \(\beta = \sum\nolimits_{k = 1}^l {{b^k}\,{\alpha _k}}\) is, say, a positive root (i.e., with coefficients b^{k} ≥ 0) belonging to Δ_{0}, then β − θ[β] = 0 is given by a sum of positive roots, weighted by nonnegative coefficients, \(\sum\nolimits_{k = 1}^r {{b^k}} ({\alpha _k}  \theta [{\alpha _k}])\). As a consequence, the coefficients b^{k} are all zero for k = 1, ⋯, r and B_{0} constitutes a basis of Δ_{0}, as claimed.
To determine completely θ we just need to know its action on a basis of simple roots. For those belonging to B_{0} it is the identity, while for the other ones we have to compute the coefficients \(a_k^j\), in Equation (6.145). These are obtained by solving the linear system given by the scalar products of these equations with the elements of B_{0},
Solving these equations for the unknown coefficients \(a_k^j\), is always possible because the Killing metric is nondegenerate on B_{0}.
The black roots of a TitsSatake diagram represent B_{0} and constitute the Dynkin diagram of the compact part \({\mathfrak m}\) of the centralizer of \({\mathfrak a}\). Because m is compact, it is the direct sum of a semisimple compact Lie algebra and onedimensional, Abelian \({{\mathfrak u}(1)}\) summands. The rank of \({\mathfrak m}\) (defined as the dimension of its maximal Abelian subalgebra; diagonalizability is automatic here because one is in the compact case) is equal to the sum of the rank of its semisimple part and of the number of \({{\mathfrak u}(1)}\) terms, while the dimension of \({\mathfrak m}\) is equal to the dimension of its semisimple part and of the number of \({{\mathfrak u}(1)}\) terms. The Dynkin diagram of \({\mathfrak m}\) reduces to the Dynkin diagram of its semisimple part.
The rank of the compact subalgebra \({\mathfrak m}\) is given by
where rank \({\mathfrak p}\), called as we have indicated above the real rank of \({\mathfrak g}\), is given by the number of cycles of the permutation π (since two simple white roots joined by a doublearrow project on the same simple restricted root [5, 93]). These two sets of data allow one to determine the dimension of \({\mathfrak m}\) (without missing \({{\mathfrak u}(1)}\) generators) [5, 93]. Another useful information, which can be directly read off from the TitsSatake diagrams is the dimension of the noncompact subspace \({\mathfrak p}\) appearing in the splitting \({\mathfrak g} = {\mathfrak k} \oplus {\mathfrak p}\). It is given (see Section 6.6.6) by
This can be illustrated in the two previous examples. For \({\mathfrak {su}}(3,\, 2)\), one gets dim \({\mathfrak g} = 24\), rank \({\mathfrak g} = 4\) and rank \({\mathfrak p} = 2\). It follows that rank \({\mathfrak m} = 2\) and since \({\mathfrak m}\) has no semisimple part (no black root), it reduces to \({\mathfrak m} = {{\mathfrak u}(1)} \oplus {\mathfrak u} (1)\) and has dimension 2. This yields dim \({\mathfrak p} = 12\), and, by substraction, \({\rm{dim}}\, {\mathfrak k} = 12\) (\({\mathfrak k}\) is easily verified to be equal to \({\mathfrak {su}}(3) \oplus {\mathfrak {su}}(2) \oplus {\mathfrak u}(1)\) Similarly, for su(4,1), one gets dim \({\rm{dim}}\, {\mathfrak g} = 24\), rank \({\mathfrak g} = 4\) and rank \({\mathfrak p} = 1\). It follows that rank \({\mathfrak m} = 3\) and since the semisimple part of \({\mathfrak m}\) is read from the black roots to be \({\mathfrak {su}}(3)\), which has rank two, one deduces \({\mathfrak m} = {\mathfrak {su}}(3) \oplus {\mathfrak u}(1)\) and \({\rm{dim}}\, {\mathfrak m} = 9\). This yields \({\rm{dim}}\, {\mathfrak p} = 8\), and, by substraction, \(\dim \,{\mathfrak k} = 16\) (\({\mathfrak k}\) is easily verified to be equal to \({\mathfrak {su}}(4) \oplus {\mathfrak u}(1)\) in this case).
Finally, from the knowledge of θ, we may obtain the restricted root space by projecting the root space according to
and restricting their action on \({\mathfrak a}\) since α and −θ(α) project on the same restricted root [5, 93].
6.6.5 Illustration: F_{4}
The Lie algebra F_{4} is a 52dimensional simple Lie algebra of rank 4. Its root vectors can be expressed in terms of the elements of an orthonormal basis {e_{k} ∣k = 1, …, 4} of a fourdimensional Euclidean space:
A basis of simple roots is
The corresponding Dynkin diagram can be obtained from Figure 37 by ignoring the painting of the vertices. To the real Lie algebra, denoted F II in [28], is associated the TitsSatake diagram of the left hand side of Figure 37. We immediately obtain from this diagram the following information:
Accordingly, F II has signature \( 21\,({\rm{compact}}\,{\rm{part}}) + ({\rm{rank}}\, {\rm{of}}\,{\mathfrak p}) =  20\) and is denoted F_{4(−20)}. Moreover, solving a system of three equations, we obtain: θ[α_{4}] = −α_{4} − α_{1} − 2 α_{2} − 3 α_{3}, i.e.,
This shows that the projection defining the reduced root system Σ consists of projecting any given root orthogonally onto its e_{1} component. Thus we obtain \(\Sigma = \{\pm {1 \over 2}{e_1},\, \pm {e_1}\}\), with multiplicity 8 for \({1 \over 2}{e_1}\) (resulting from the projection of the eight roots \(\{{1 \over 2}({e_1} \pm {e_2} \pm {e_3} \pm {e_4})\}\) and 7 for e_{1} (resulting from the projection of the seven roots {e_{1} ± e_{k} ∣ k = 2, 3, 4} ∪ {e_{1}}).
Let us mention that, contrary to the Vogan diagrams, any “formal TitsSatake diagram” is not admissible. For instance if we consider the right hand side diagram of Figure 37 we get
But this means that for the root α = e_{1}, α + θ*[α] = e_{1} − e_{2} is again a root, which is impossible as we shall see below.
6.6.6 Some more formal considerations
Let us recall some crucial aspects of the discussion so far. Let \({{\mathfrak g}_\sigma}\) be a real form of the complex semisimple Lie algebra \({{\mathfrak g}^{\mathbb C}}\) and σ be the conjugation it defines. We have seen that there always exists a compact real Lie algebra \({{\mathfrak u}_\tau}\) such that the corresponding conjugation τ commutes with σ. Moreover, we may choose a Cartan subalgebra \({\mathfrak h}\) of \({{\mathfrak u}_\tau}\) such that its complexification \({{\mathfrak h}^{\mathbb C}}\) is invariant under σ, i.e., \(\sigma ({{\mathfrak h}^{\mathbb C}}) = {{\mathfrak h}^{\mathbb C}}\). Then the real form \({{\mathfrak g}_\sigma}\) is said to be normally related to \(({\mathfrak u}_\theta,\, {\mathfrak h})\). As previously, we denote by the same letter θ the involution defined by duality on \(({\mathfrak h}^{\mathbb C}){\ast}\) (and also on the root lattice with respect to \({\mathfrak h}^{\mathbb C}: {\Delta}\)) by θ = τσ.
When \({{\mathfrak g}_\sigma}\) and \({{\mathfrak u}_\tau}\) are normally related, we may decompose the former into compact and noncompact components \({\mathfrak g}_{\sigma} = {\mathfrak k} \oplus {\mathfrak p}\) such that \({{\mathfrak u}_\tau} = {\mathfrak k} \oplus i{\mathfrak p}\). As mentioned, the starting point consists of choosing a maximally Abelian noncompact subalgebra \({\mathfrak a} \subset {\mathfrak p}\) and extending it to a Cartan subalgebra \({\mathfrak h} = {\mathfrak t} \oplus {\mathfrak a}\), where \({\mathfrak t} \subset {\mathfrak k}\). This Cartan subalgebra allows one to consider the real Cartan subalgebra
Let us remind the reader that, in this case, the Cartan involution θ = στ = τσ is such that \(\theta {\vert_{\mathfrak k}} = + 1\) and \(\theta {\vert_{\mathfrak p}} =  1\). From Equation (6.69) we obtain
and using θ^{2} = 1 we deduce that
Furthermore, Equation (6.32) and the fact that the structure constants are rational yield the following relations:
On the other hand, the commutativity of τ and σ implies
with
In particular, if the root α belongs to Δ_{0}, defined in Equation (6.146), then θ[α] = α and thus \(\rho _\alpha ^2 = 1\), i.e.,
Let us denote by Δ_{0}, and Δ_{0,+} the subsets of Δ_{0} corresponding to the imaginary noncompact and imaginary compact roots, respectively. We have
Obviously, for α ∈ Δ_{0, −}, E_{α} belongs to \({{\mathfrak p}^{\mathbb C}}\), while for α ∈ Δ_{0, +}, E_{α} belongs to \({{\mathfrak k}^{\mathbb C}}\). Moreover, if α ∈ Δ Δ_{0} we find
These remarks lead to the following explicit constructions of the complexifications of \({\mathfrak k}\) and \({\mathfrak p}\):
Furthermore, since θ fixes all the elements of Δ_{0}, the subspace \({\oplus _{\alpha \in {\Delta _{0,\, }}}}\,{\mathbb C}{E_\alpha}\) belongs to the centralizer^{Footnote 29} of \({\mathfrak a}\) and thus is empty if \({\mathfrak a}\) is maximally Abelian in \({\mathfrak p}\). Taking this remark into account, we immediately obtain the dimension formulas (6.149, 6.150).
Using, as before, the basis in Equation (6.147) we obtain for the roots belonging to B B_{0}, i.e., for an index i ≤ r:
Thus
As \(\sum\nolimits_{j = 1,\, \ldots, \, r} {p_i^jp_j^k = \delta _i^j}\), where the coefficients \({p_i^j}\) are nonnegative integers, the matrix \(({p_i^j})\) must be a permutation matrix and it follows that
where π is an involutive permutation of {1, …, r}.
A fundamental property of Δ is
To show this, note that if α ∈ Δ_{0}, it would imply that 2 α belongs to Δ, which is impossible for the root lattice of a semisimple Lie algebra. If α ∈ Δ Δ_{0} and θ[α] + α ∈ Δ, then θ[α] + α ∈ Δ_{0}. Thus we obtain using Equation (6.35) and taking into account that \({\mathfrak a}\) is maximal Abelian in \({\mathfrak p}\), that ρ_{α} = +1, i.e.,
and
From this result we deduce
i.e., \({\rho _\alpha} =  \overline {\rho  \alpha}\) which is incompatible with equation (6.160). Thus, the statement (6.170) follows.
6.7 The real semisimple algebras \(\mathfrak{so}(k,\,l)\)
The dimensional reduction from 10 to 3 dimensions of \({\mathcal N} = 1\) supergravity coupled to m Maxwell multiplets leads to a nonlinear sigma model \({\mathcal G}/{\mathcal K}({\mathcal G})\) with \(\mathrm{Lie}(\mathcal{G})=\mathfrak{so}(8,\, 8+m)\) (see Section 7). To investigate the geometry of these cosets, we shall construct their TitsSatake diagrams.
The \(\mathfrak{so}(n,\, \mathbb{C})\) Lie algebra can be represented by n × n antisymmetric complex matrices. The compact real form is \(\mathfrak{so}(k+l,\, \mathbb{R})\), naturally represented as the set of n × n antisymmetric real matrices. One way to describe the real subalgebras \(\mathfrak{so}(k,\, l)\), aligned with the compact form \(\mathfrak{so}(k+l,\, \mathbb{R})\), is to consider \(\mathfrak{so}(k,\, l)\) as the set of infinitesimal rotations expressed in Pauli coordinates, i.e., to represent the hyperbolic space on which they act as a Euclidean space whose first k coordinates, x^{a}, are real while the last l coordinates y^{b} are purely imaginary. Writing the matrices of \(\mathfrak{so}(k,\, l)\) in block form as
where
we may obtain a maximal Abelian subspace \(\mathfrak{a}\) by allowing C to have nonzero elements only on its diagonal, i.e., to be of the form:
with k > l or l < k, respectively.
To proceed, let us denote by H_{j} the matrices whose entries are everywhere vanishing except for a 2 × 2 block,
, on the diagonal. These matrices have the following realisation in terms of the \({K^i}_j\) (defined in Equation (6.83)):
They constitute a set of \(\mathfrak{so}(k + l)\) commuting generators that provide a Cartan subalgebra; it will be the Cartan subalgebra fixed by the Cartan involution defined by the real forms that we shall now discuss.
6.7.1 Dimensions l = 2q +1 < k = 2p
Motivated by the dimensional reduction of supergravity, we shall assume k = 2p, even. We first consider l = 2q + 1 < k. Then by reordering the coordinates as follows,
we obtain a Cartan subalgebra of \(\mathfrak{so}(2q + 1,\,2p)\), with noncompact generators first, and aligned with the one introduced in Equation (6.177) by considering the basis {i H_{1}, ⋯, i H_{l}, H_{l+1}, ⋯, H_{q+p}}^{Footnote 30}. These generators are all orthogonal to each other. Let us denote the elements of the dual basis by {f_{A} ∣ A = 1, ⋯, p + q}, and split them into two subsets: {f_{a} ∣ a = 1, ⋯, 2q + 1} and {f_{α} ∣ α = 2q + 2, ⋯, p + q}. The action of the Cartan involution on these generators is very simple,
The root system of \(\mathfrak{so}(2q + 1,\,2p)\) is B_{(p+q)}, represented by Δ = {±f_{A} ± f_{B} ∣ A < B = 1, ⋯, p + q} ∪ {±f_{A} ∣ A1, ⋯, p + q}. A simple root basis can be taken as:
It is then straigthforward to obtain the action of the Cartan involution on the simple roots:
The corresponding TitsSatake diagrams are displayed in Figure 38.
From Equation (6.179) we also obtain without effort that the set of restricted roots consists of the 4q(2q + 1) roots {±f_{a} ± f_{b}}, each of multiplicity one, and the 4q + 2 roots {±f_{a}}, each of multiplicity 2(p − q) − 1. These constitute a B_{2q+1} root system.
6.7.2 Dimensions l = 2q + 1 > k = 2p
Following the same procedure as for the previous case, we obtain a Cartan subalgebra consisting of 2p noncompact generators and q − p compact generators. The corresponding TitsSatake diagrams are displayed in Figure 39.
The restricted root system is now of type B_{2p}, with 4p(2p − 1) long roots of multiplicity one and 4p short roots of multiplicity 2(q − p) + 1.
6.7.3 Dimensions l = 2q, k = 2p
Here the root system is of type D_{p+q}, represented by Δ = {±f_{A} ± f_{B} ∣ A < B = 1, ⋯, p + q}, where the orthonormal vectors f_{A} again constitute a basis dual to the natural Cartan subalgebra of \(\mathfrak{so}(k+l)\). Now, k = 2p and l = 2q are both assumed even, and we may always suppose k ≥ l. The Cartan involution to be considered acts as previously on the f_{A}:
and
The simple roots can be chosen as
on which the Cartan involution has the following action:
For q = p
$$\theta [{\alpha _A}] =  {\alpha _A}\qquad {\rm{for}}\,A = 1,\, \cdots ,\,q + p.$$(6.182)For q = p − 1
$$\begin{array}{*{20}c} {\theta [{\alpha _A}] =  {\alpha _A}\quad \quad \quad \quad {\rm{for}}\,A = 1,\, \cdots ,\,2q = q + p  1,} \\ {\theta [{\alpha _{q + p  1}}] =  {\alpha _{q + p}},\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\theta [{\alpha _{q + p}}] =  {\alpha _{q + p  1}}.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$(6.183)For q < p − 1
$$\begin{array}{*{20}c} {\theta [{\alpha _A}] =  {\alpha _A}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad A = 1,\, \cdots ,\,\,2q  1,\quad \quad \quad \quad} \\ {\theta [{\alpha _{2q}}] =  {\alpha _{2q}}  2({\alpha _{2q + 1}} + \cdots ,{\alpha _{q + p  2}})\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ { {\alpha _{q + p  1}}  {\alpha _{q + p}},\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\theta [{\alpha _A}] = + {\alpha _A}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad A = 2q + 1, \cdots ,\,q + p,\quad \quad \quad} \\ \end{array}$$(6.184)
The corresponding TitsSatake diagrams are obtained in the same way as before and are displayed in Figure 40
When q < p, the restricted root system is again of type B2q, with 4q(2q − 1) long roots of multiplicity one and 4q short roots of multiplicity 2(p − q). For p = q, the short roots disappear and the restricted root system is of D2p type, with all roots having multiplicity one.
6.8 Summary — TitsSatake diagrams for noncompact real forms
To summarize the analysis, we provide the TitsSatake diagrams for all noncompact real forms of all simple Lie algebras [5, 93]. We do not give explicitly the TitsSatake diagrams of the compact real forms as these are simply obtained by painting in black all the roots of the standard Dynkin diagrams.
Theorem: The simple real Lie algebras are:
The Lie algebras \(\mathfrak{g}^{\mathbb{R}}\) where \(\mathfrak{g}\) is one of the complex simple Lie algebras A_{n} (n ≥ 1), B_{n} (n ≥ 2), C_{n} (n ≥ 3), D_{n} (n ≥ 4), G_{2}, F_{4}, E_{6}, E_{7}, or E_{8}, and the compact real forms of these.
The classical real Lie algebras of types \(\mathfrak{su},\,\mathfrak{so},\,\mathfrak{sp}\) and \(\mathfrak{sl}\). These are listed in Table 26.
The twelve exceptional real Lie algebras, listed in Table 27 (our conventions are due to Cartan).
7 KacMoody Billiards II — The Case of NonSplit Real Forms
We will now make use of the results from the previous section to extend the analysis of KacMoody billiards to include also theories whose Uduality symmetries are described by algebras \(\mathfrak{u}_3\) that are nonsplit. The key concepts are that of restricted root systems, restricted Weyl group — and the associated concept of maximal split subalgebra — as well as the Iwasawa decomposition already encountered above. These play a prominent role in our discussion as they determine the billiard structure. We mainly follow [95].
7.1 The restricted Weyl group and the maximal split “subalgebra”
Let \(\mathfrak{u}_3\) be any real form of the complex Lie algebra \(\mathfrak{u}_3^{\mathbb{C}},\,\theta\), its Cartan involution, and let
be the corresponding Cartan decomposition. Furthermore, let
be a maximal noncompact Cartan subalgebra, with \(\mathfrak{t}_3\) (respectively, \(\mathfrak{a}_3\)) its compact (respectively, noncompact) part. The real rank of \(\mathfrak{u}_3\) is, as we have seen, the dimension of \(\mathfrak{a}_3\). Let now Δ denote the root system of \(\mathfrak{u}_{3}^{\mathbb{C}},\,\Sigma\), the restricted root system and m_{λ} the multiplicity of the restricted root λ.
As explained in Section 4.9.2, the restricted root system of the real form \(\mathfrak{u}_{3}\) can be either reduced or nonreduced. If it is reduced, it corresponds to one of the root systems of the finitedimensional simple Lie algebras. On the other hand, if the restricted root system is nonreduced, it is necessarily of (BC)_{n}type [93] (see Figure 19 for a graphical presentation of the BC_{3} root system).
7.1.1 The restricted Weyl group
By definition, the restricted Weyl group is the Coxeter group generated by the fundamental reflections, Equation (4.55), with respect to the simple roots of the restricted root system. The restricted Weyl group preserves multiplicities [93].
7.1.2 The maximal split “subalgebra” \(\mathfrak{f}\)
Although multiplicities are an essential ingredient for describing the full symmetry \(\mathfrak{u}_3\), they turn out to be irrelevant for the construction of the gravitational billiard. For this reason, it is useful to consider the maximal split “subalgebra” \(\mathfrak{f}\), which is defined as the real, semisimple, split Lie algebra with the same root system as the restricted root system as \(\mathfrak{u}_3\) (in the (BC)_{n}case, we choose for definiteness the root system of \(\mathfrak{f}\) to be of B_{n}type). The real rank of \(\mathfrak{f}\) coincides with the rank of its complexification \(\mathfrak{f}^{\mathbb{C}}\), and one can find a Cartan subalgebra \(\mathfrak{h}_{\mathfrak{f}}\) of \(\mathfrak{f}\), consisting of all generators of \(\mathfrak{h}_3\) which are diagonalizable over the reals. This subalgebra \(\mathfrak{h}_{\mathfrak{f}}\) has the same dimension as the maximal noncompact subalgebra \(\mathfrak{a}_3\) of the Cartan subalgebra \(\mathfrak{h}_3\) of \(\mathfrak{u}_3\).
By construction, the root space decomposition of \(\mathfrak{f}\) with respect to \(\mathfrak{h}_{\mathfrak{f}}\) provides the same root system as the restricted root space decomposition of \(\mathfrak{u}_3\) with respect to \(\mathfrak{a}_3\), except for multiplicities, which are all trivial (i.e., equal to one) for \(\mathfrak{f}\). In the (BC)_{n}case, there is also the possibility that twice a root of \(\mathfrak{f}\) might be a root of \(\mathfrak{u}_3\). It is only when \(\mathfrak{u}_3\) is itself split that \(\mathfrak{f}\) and \(\mathfrak{u}_3\) coincide.
One calls \(\mathfrak{f}\) the “split symmetry algebra”. It contains as we shall see all the information about the billiard region [95]. How \(\mathfrak{f}\) can be embedded as a subalgebra of \(\mathfrak{u}_3\) is not a question that shall be of our concern here.
7.1.3 The Iwasawa decomposition and scalar coset Lagrangians
The purpose of this section is to use the Iwasawa decomposition for \(\mathfrak{u}_3\) described in Section 6.4.5 to derive the scalar Lagrangian based on the coset space \({{\mathcal U}_3}/{\mathcal K}({{\mathcal U}_3})\). The important point is to understand the origin of the similarities between the two Lagrangians in Equation (5.45) and Equation (7.8) below.
The full algebra \(\mathfrak{u}_3\) is subject to the root space decomposition
with respect to the restricted root system. For each restricted root λ, the space g_{λ} has dimension m_{λ}. The nilpotent algebra \(\mathfrak{n}_{3} \subset \mathfrak{u}_3\), consisting of positive root generators only, is the direct sum
over positive roots. The Iwasawa decomposition of the Uduality algebra \(\mathfrak{u}_3\) reads
(see Section 6.4.5). It is \(\mathfrak{a}_3\) that appears in Equation (7.5) and not the full Cartan subalgebra \(\mathfrak{h}_3\) since the compact part of \(\mathfrak{h}_3\) belongs to \(\mathfrak{k}_3\).
This implies that when constructing a Lagrangian based on the coset space \({{\mathcal U}_3}/{\mathcal K}({{\mathcal U}_3})\), the only part of \(\mathfrak{u}_3\) that will show up in the Borel gauge is the Borel subalgebra
Thus, there will be a number of dilatons equal to the dimension of \(\mathfrak{a}_3\), i.e., equal to the real rank of \(\mathfrak{u}_3\), and axion fields for the restricted roots (with multiplicities).
More specifically, an (xdependent) element of the coset space \({{\mathcal U}_3}/{\mathcal K}({{\mathcal U}_3})\) takes the form
where the dilatons ϕ and the axions χ are coordinates on the coset space, and where x denotes an arbitrary set of parameters on which the coset element might depend. The corresponding Lagrangian becomes
where the sums over s_{α} = 1, ⋯, mult α are sums over the multiplicities of the positive restricted roots α.
By comparing Equation (7.8) with the corresponding expression (5.45) for the split case, it is clear why it is the maximal split subalgebra of the Uduality algebra that is relevant for the gravitational billiard. Were it not for the additional sum over multiplicities, Equation (7.8) would exactly be the Lagrangian for the coset space \({\mathcal F}/{\mathcal K}({\mathcal F})\), where \(\mathfrak{k}_{\mathfrak{f}}=\mathrm{Lie}\, \mathcal{K}(\mathcal{F})\) is the maximal compact subalgebra of \(\mathfrak{f}\) (note that \(\mathfrak{k}_{\mathfrak{f}}\neq \mathfrak{k}_{3}\)). Recall now that from the point of view of the billiard, the positive roots correspond to walls that deflect the particle motion in the Cartan subalgebra. Therefore, multiplicities of roots are irrelevant since these will only result in several walls stacked on top of each other without affecting the dynamics. (In the (BC)_{n}case, the wall associated with 2λ is furthermore subdominant with respect to the wall associated with λ when both λ and 2λ are restricted roots, so one can keep only the wall associated with λ. This follows from the fact that in the (BC)_{n}case the root system of \(\mathfrak{f}\) is taken to be of B_{n}type.)
7.2 “Split symmetry controls chaos”
The main point of this section is to illustrate and explain the statement “split symmetry controls chaos” [95]. To this end, we will now extend the analysis of Section 5 to nonsplit real forms, using the Iwasawa decomposition. As we have seen, there are two main cases to be considered:
The restricted root system Σ of u_{3} is of reduced type, in which case it is one of the standard root systems for the Lie algebras A_{n}, B_{n}, C_{n}, D_{n}, G_{2}, F_{4}, E_{6}, E_{7} or E_{8}.
The restricted root system, Σ, of \(\mathfrak{u}_3\) is of nonreduced type, in which case it is of (BC)_{n}type.
In the first case, the billiard is governed by the overextended algebra \(\mathfrak{f}^{++}\), where \(\mathfrak{f}\) is the “maximal split subalgebra” of \(\mathfrak{u}_3\). Indeed, the coupling to gravity of the coset Lagrangian of Equation (7.8) will introduce, besides the simple roots of \(\mathfrak{f}\) (electric walls) the affine root of \(\mathfrak{f}\) (dominant magnetic wall) and the overextended root (symmetry wall), just as in the split case (but for \(\mathfrak{f}\) instead of \(\mathfrak{u}_3\)). This is therefore a straightforward generalization of the analysis in Section 5.
The second case, however, introduces a new phenomenon, the twisted overextensions of Section 4. This is because the highest root of the (BC)_{n} system differs from the highest root of the B_{n} system. Hence, the dominant magnetic wall will provide a twisted affine root, to which the symmetry wall will attach itself as usual [95].
We illustrate the two possible cases in terms of explicit examples. The first one is the simplest case for which a twisted overextension appears, namely the case of pure fourdimensional gravity coupled to a Maxwell field. This is the bosonic sector of \({\mathcal N} = 2\) supergravity in four dimensions, which has the nonsplit real form \(\mathfrak{su}(2,\,1)\) as its Uduality symmetry. The restricted root system of \(\mathfrak{su}(2,\,1)\) is the nonreduced (BC)_{1}system, and, consequently, as we shall see explicitly, the billiard is governed by the twisted overextension \(A_2^{(2) +}\).
The second example is that of heterotic supergravity, which exhibits an SO(8, 24)/(SO(8) × SO(24)) coset symmetry in three dimensions. The Uduality algebra is thus \(\mathfrak{so}(8,\,24)\), which is nonsplit. In this example, however, the restricted root system is B_{8}, which is reduced, and so the billiard is governed by a standard overextension of the maximal split subalgebra \(\mathfrak{so}(8,\,9)\subset \mathfrak{so}(8,\,24)\).
7.2.1 (BC)_{1} and \({\mathcal N} = 2,\,D = 4\) pure supergravity
We consider \({\mathcal N} = 2\) supergravity in four dimensions where the bosonic sector consists of gravity coupled to a Maxwell field. It is illuminating to compare the construction of the billiard in the two limiting dimensions, D = 4 and D = 3.
In maximal dimension the metric contains three scale factors, β^{1}, β^{2} and β^{3}, which give rise to three symmetry wall forms,
where only s_{21} and s_{32} are dominant. In four dimensions the curvature walls read
Finally we have the electric and magnetic wall forms of the Maxwell field. These are equal because there is no dilaton. Hence, the wall forms are
The billiard region \({{\mathcal B}_{{{\mathcal M}_\beta}}}\) is defined by the set of dominant wall forms,
The first dominant wall form, e_{1} (β), is twice degenerate because it occurs once as an electric wall form and once as a magnetic wall form. Because of the existence of the curvature wall, c_{1}(β) = 2β^{1}, we see that 2α_{1} is also a root.
The same billiard emerges after reduction to three spacetime dimensions, where the algebraic structure is easier to exhibit. As before, we perform the reduction along the first spatial direction. The associated scale factor is then replaced by the KaluzaKlein dilaton \(\hat \varphi\) such that
The remaining scale factors change accordingly,
and the two symmetry walls become
In addition to the dilaton \(\hat \varphi\), there are three axions: one \((\hat \chi)\) arising from the dualization of the KaluzaKlein vector, one \(({\hat \chi ^E})\) coming from the component A_{1} of the Maxwell vector potential and one \(({\hat \chi ^C})\) coming from dualization of the Maxwell vector potential in 3 dimensions (see, e.g., [35] for a review). There are then a total of four scalars. These parametrize the coset space SU(2, 1)/S(U(2) × U(1)) [113].
The EinsteinMaxwell Lagrangian in four dimensions yields indeed in three dimensions the Einsteinscalar Lagrangian, where the Lagrangian for the scalar fields is given by
with
Here, the ellipses denotes terms that are not relevant for understanding the billiard structure. The Uduality algebra of \({\mathcal N} = 2\) supergravity compactified to three dimensions is therefore
which is a nonsplit real form of the complex Lie algebra \(\mathfrak{sl}(3,\,\mathbb{C})\). This is in agreement with Table 1 of [113]. The restricted root system of \(\mathfrak{su}(2,\,1)\) is of (BC)_{1}type (see Table 28 in Section 6.8) and has four roots: α_{1}, 2α_{1}, −α_{1} and −2α_{1}. One may take α_{1} to be the simple root, in which case Σ_{+} = {α_{1}, 2α_{1}} and 2α_{1} is the highest root. The short root α_{1} is degenerate twice while the long root 2α_{1} is nondegenerate. The Lagrangian (7.16) coincides with the Lagrangian (7.8) for \(\mathfrak{su}(2,\,1)\) with the identification
We clearly see from the Lagrangian that the simple root \({\hat \alpha _1}\) has multiplicity 2 in the restricted root system, since the corresponding wall appears twice. The maximal split subalgebra may be taken to be \(A_{1}\equiv\mathfrak{su}(1,\,1)\) with root system \(\{{\hat \alpha _1},\,  {\hat \alpha _1}\}\).
Let us now see how one goes from \(\mathfrak{su}(2,\,1)\) described by the scalar Lagrangian to the full algebra, by including the gravitational scale factors. Let us examine in particular how the twist arises. For the standard root system of A_{1} the highest root is just \({\hat \alpha _1}\). However, as we have seen, for the (BC)_{1} root system the highest root is \({\theta _{{{(BC)}_1}}} = 2{\hat \alpha _1}\), with
So we see that because of \(({\hat \alpha _1}\vert{\hat \alpha _1}) = {1 \over 2}\), the highest root \({\theta _{{{(BC)}_1}}}\) already comes with the desired normalization. The affine root is therefore
whose norm is
The scalar product between \({\hat \alpha _1}\) and \({\hat \alpha _2}\) is \(({\hat \alpha _1}\vert{\hat \alpha _2}) =  1\) and the Cartan matrix at this stage becomes (i, j = 1, 2)
which may be identified not with the affine extension of A_{1} but with the Cartan matrix of the twisted affine KacMoody algebra \(A_2^{(2)}\). It is the underlying (BC)_{1} root system that is solely responsible for the appearance of the twist. Because of the fact that \({\theta _{{{(BC)}_1}}} = 2{\hat \alpha _1}\) the two simple roots of the affine extension come with different length and hence the asymmetric Cartan matrix in Equation (7.22). It remains to include the overextended root
which has nonvanishing scalar product only with \({\hat \alpha _2},\,({\hat \alpha _2}\vert{\hat \alpha _3}) =  1\), and so its node in the Dynkin diagram is attached to the second node by a single link. The complete Cartan matrix is
which is the Cartan matrix of the Lorentzian extension \(A_2^{(2) +}\) of \(A_2^{(2)}\) henceforth referred to as the twisted overextension of A_{1}. Its Dynkin diagram is displayed in Figure 41.
The algebra \(A_2^{(2) +}\) was already analyzed in Section 4, where it was shown that its Weyl group coincides with the Weyl group of the algebra \(A_1^{+ +}\). Thus, in the BKLlimit the dynamics of the coupled EinsteinMaxwell system in fourdimensions is equivalent to that of pure fourdimensional gravity, although the set of dominant walls are different. Both theories are chaotic.
7.2.2 Heterotic supergravity and \({\mathfrak {sl}(8,\,24)}\)
Pure \({\mathcal N} = 1\) supergravity in D = 10 dimensions has a billiard description in terms of the hyperbolic KacMoody algebra \(D{E_{10}} = D_8^{+ +}\) [45]. This algebra is the overextension of the Uduality algebra, \({{\mathfrak u}_3} = {D_8} = {\mathfrak {so}}(8,\,8)\), appearing upon compactification to three dimensions. In this case, \({\mathfrak {so}}(8,\,8)\) is the split form of the complex Lie algebra D_{8}, so we have \({\mathfrak f} = {{\mathfrak u}_3}\).
By adding one Maxwell field to the theory we modify the billiard to the hyperbolic KacMoody algebra \(B{E_{10}} = B_8^{+ +}\), which is the overextension of the split form \({\mathfrak {so}}(8,\,9)\) of B_{8} [45]. This is the case relevant for (the bosonic sector of) Type I supergravity in ten dimensions. In both these cases the relevant KacMoody algebra is the overextension of a split real form and so falls under the classification given in Section 5.
Let us now consider an interesting example for which the relevant Uduality algebra is nonsplit. For the heterotic string, the bosonic field content of the corresponding supergravity is given by pure gravity coupled to a dilaton, a 2form and an E_{8} × E_{8} YangMills gauge field. Assuming the gauge field to be in the Cartan subalgebra, this amounts to adding 16 \({\mathcal N} = 1\) vector multiplets in the bosonic sector, i.e., to adding 16 Maxwell fields to the tendimensional theory discussed above. Geometrically, these 16 Maxwell fields correspond to the KaluzaKlein vectors arising from the compactification on T^{16} of the 26dimensional bosonic leftmoving sector of the heterotic string [89].
Consequently, the relevant Uduality algebra is \({\mathfrak {so}}(8,\,8+16)={\mathfrak {so}}(8,\,24)\) which is a nonsplit real form. But we know that the billiard for the heterotic string is governed by the same KacMoody algebra as for the Type I case mentioned above, namely \(B{E_{10}} \equiv {\mathfrak {so}}{(8,\,9)^{+ +}}\), and not \({\mathfrak {so}}(8,\,24)^{+ +}\) as one might have expected [45]. The only difference is that the walls associated with the oneforms are degenerate 16 times. We now want to understand this apparent discrepancy using the machinery of nonsplit real forms exhibited in previous sections. The same discussion applies to the SO(32)superstring.
In three dimensions the heterotic supergravity Lagrangian is given by a pure threedimensional EinsteinHilbert term coupled to a nonlinear sigma model for the coset SO(8, 24)/(SO(8) × SO(24)). This Lagrangian can be understood by analyzing the Iwasawa decomposition of \({\mathfrak {so}}(8,\,24) = {\rm{Lie}}[SO(8,\,24)]\). The maximal compact subalgebra is
This subalgebra does not appear in the sigma model since it is divided out in the coset construction (see Equation (7.7)) and hence we only need to investigate the Borel subalgebra \({\mathfrak {a}_3} \oplus {\mathfrak {n}_3}\) of \({\mathfrak {so}}(8,\,24)\) in more detail.
As was emphasized in Section 7.1, an important feature of the Iwasawa decomposition is that the full Cartan subalgebra \({{\mathfrak h}_3}\) does not appear explicitly but only the maximal noncompact Cartan subalgebra \({{\mathfrak a}_3}\), associated with the restricted root system. This is the maximal Abelian subalgebra of \({\mathfrak {u}_3} = {\mathfrak {so}}(8,\,24)\), whose adjoint action can be diagonalized over the reals. The remaining Cartan generators of \({{\mathfrak h}_3}\) are compact and so their adjoint actions have imaginary eigenvalues. The general case of \({\mathfrak {so}}(2q,\,2p)\) was analyzed in detail in Section 6.7 where it was found that if q < p, the restricted root system is of type B_{2q}. For the case at hand we have q = 4 and p = 12 which implies that the restricted root system of \({\mathfrak {so}}(8,\,24)\) is (modulo multiplicities) \({\Sigma _{{\mathfrak {so}}(8,\,24)}} = {B_8}\).
The root system of B_{8} is eightdimensional and hence there are eight Cartan generators that may be simultaneously diagonalized over the real numbers. Therefore the real rank of \({\mathfrak {so}}(8,\,24)\) is eight, i.e.,
Moreover, it was shown in Section 6.7 that the restricted root system of \({\mathfrak {so}}(2p,\,2q)\) has 4q(2q − 1) long roots which are nondegenerate, i.e., with multiplicity one, and 4q long roots with multiplicities 2(p − q). In the example under consideration this corresponds to seven nondegenerate simple roots α_{1}, ⋯, α_{7} and one short simple root α_{8} with multiplicity 16. The Dynkin diagram for the restricted root system \({\Sigma _{{\mathfrak {so}}(8,\,24)}}\) is displayed in Figure 42 with the multiplicity indicated in brackets over the short root. It is important to note that the restricted root system \({\mathfrak {so}}(8,\,24)\) differs from the standard root system of \({\mathfrak {so}}(8,\,9)\) precisely because of the multiplicity 16 of the simple root α_{8}.
Because of these properties of \({\mathfrak {so}}(8,\,24)\) the Lagrangian for the coset
takes a form very similar to the Lagrangian for the coset
The algebra constructed from the restricted root system B_{8} is the maximal split subalgebra
Let us now take a closer look at the Lagrangian in three spacetime dimensions We parametrize an element of the coset by
where x^{μ} (μ = 0, 1, 2) are the coordinates of the external threedimensional spacetime, \(\alpha _i^ \vee\) are the noncompact Cartan generators and Δ_{+} denotes the full set of positive roots of \({\mathfrak {so}}(8,\,24)\).
The Lagrangian constructed from the coset representative in Equation (7.30) becomes (again, neglecting corrections to the single derivative terms of the form “∂_{x}χ”
where \({{\tilde \Sigma}^ +}\) denotes all nonsimple positive roots of Σ, i.e.,
with
This Lagrangian is equivalent to the Lagrangian for SO(8, 9)/(SO(8) × SO(9)) except for the existence of the nontrivial root multiplicities.
The billiard for this theory can now be computed with the same methods that were treated in detail in Section 5.3.3. In the BKLlimit, the simple roots α_{1}, ⋯, α_{8} become the nongravitational dominant wall forms. In addition to this we get one magnetic and one gravitational dominant wall form:
where θ(ϕ) is the highest root of \({\mathfrak {so}}(8,\,9)\):
The affine root α_{0} attaches with a single link to the second simple root α_{2} in the Dynkin diagram of B_{8}. Similarly the overextended root α_{−1} attaches to α_{0} with a single link so that the resulting Dynkin diagram corresponds to BE_{10} (see Figure 43). It is important to note that the underlying root system is still an overextension of the restricted root system and hence the multiplicity of the simple short root α_{8} remains equal to 16. Of course, this does not affect the dynamics in the BKLlimit because the multiplicity of α8 simply translates to having multiple electric walls on top of each other and this does not alter the billiard motion.
This analysis again showed explicitly how it is always the split symmetry that controls the chaotic behavior in the BKLlimit. It is important to point out that when going beyond the strict BKLlimit, one expects more and more roots of the algebra to play a role. Then it is no longer sufficient to study only the maximal split subalgebra \({\mathfrak {so}}{(8,\,9)^{+ +}}\) but instead the symmetry of the theory is believed to contain the full algebra \({\mathfrak {so}}{(8,\,24)^{+ +}}\). In the spirit of [47] one may then conjecture that the dynamics of the heterotic supergravity should be equivalent to a null geodesic on the coset space \(SO{(8,\,24)^{+ +}}/{\mathcal K}(SO{(8,\,24)^{+ +}})\)) [42].
7.3 Models associated with nonsplit real forms
In this section we provide a list of all theories coupled to gravity which, upon compactification to three dimensions, display Uduality algebras that are not maximal split [95]. This therefore completes the classification of Section 5.
One can classify the various theories through the number \({\mathcal N}\) of supersymmetries that they possess in D = 4 spacetime dimensions. All pforms can be dualized to scalars or to 1forms in four dimensions so the theories all take the form of pure supergravities coupled to collections of Maxwell multiplets. The analysis performed for the split forms in Section 5.3 were all concerned with the cases of \({\mathcal N} = 0\) or \({\mathcal N} = 8\) supergravity in D = 4. We consider all pure fourdimensional supergravities \(({\mathcal N} = 1,\, \cdots, \,8)\) as well as pure \({\mathcal N} = 4\) supergravity coupled to k Maxwell multiplets.
As we have pointed out, the main new feature in the nonsplit cases is the possible appearance of socalled twisted overextensions. These arise when the restricted root system of \({\mathcal U}_3\) is of nonreduced type hence yielding a twisted affine KacMoody algebra in the affine extension of \({\mathfrak f} \subset {{\mathfrak u}_3}\). It turns out that the only cases for which the restricted root system is of nonreduced ((BC)type) is for the pure \({\mathcal N} = 2,\,3\) and \({\mathcal N} = 5\) supergravities. The example of \({\mathcal N} = 2\) was already discussed in detail before, where it was found that the Uduality algebra is \({{\mathfrak u}_3} = {\mathfrak su}(2,\,1)\) whose restricted root system is (BC)_{1}, thus giving rise to the twisted overextension \(A_2^{(2) +}\). It turns out that for the \({\mathcal N} = 3\) case the same twisted overextension appears. This is due to the fact that the Uduality algebra is \({{\mathfrak u}_3} = {\mathfrak su}(4,\,1)\) which has the same restricted root system as \({\mathfrak {so}}(2,\,1)\), namely (BC)_{1}. Hence, \(A_1^{(2) +}\) controls the BKLlimit also for this theory.
The case of \({\mathcal N} = 5\) follows along similar lines. In D = 3 the nonsplit form E_{6(−14)} of E_{6} appears, whose maximal split subalgebra is \({\mathfrak f} = {C_2}\). However, the relevant KacMoody algebra is not \(C_2^{+ +}\) but rather \(A_4^{(2) +}\) because the restricted root system of E_{6(−14)} is (BC)_{2}.
In Table 37 we display the algebraic structure for all pure supergravities in four dimensions as well as for \({\mathcal N} = 0\) supergravity with k Maxwell multiplets. We give the relevant Uduality algebras \({{\mathfrak u}_3}\), the restricted root systems Σ, the maximal split subalgebras \({\mathfrak f}\) and, finally, the resulting overextended KacMoody algebras \({\mathfrak g}\).
Let us end this section by noting that the study of real forms of hyperbolic KacMoody algebras has been pursued in [17].
8 Level Decomposition in Terms of Finite Regular Subalgebras
We have shown in the previous sections that Weyl groups of Lorentzian KacMoody algebras naturally emerge when analysing gravity in the extreme BKL regime. This has led to the conjecture that the corresponding KacMoody algebra is in fact a symmetry of the theory (most probably enlarged with new fields) [46]. The idea is that the BKL analysis is only the “revelator” of that huge symmetry, which would exist independently of that limit, without making the BKL truncations. Thus, if this conjecture is true, there should be a way to rewrite the gravity Lagrangians in such a way that the KacMoody symmetry is manifest. This conjecture itself was made previously (in this form or in similar ones) by other authors on the basis of different considerations [