Covariant homogeneous nets of standard subspaces

Rindler wedges are fundamental localization regions in AQFT. They are determined by the one-parameter group of boost symmetries fixing the wedge. The algebraic canonical construction of the free field provided by Brunetti-Guido-Longo (BGL) arises from the wedge-boost identification, the BW property and the PCT Theorem. In this paper we generalize this picture in the following way. Firstly, given a $\mathbb Z_2$-graded Lie group we define a (twisted-)local poset of abstract wedge regions. We classify (semisimple) Lie algebras supporting abstract wedges and study special wedge configurations. This allows us to exhibit an analog of the Haag-Kastler one-particle net axioms for such general Lie groups without referring to any specific spacetime. This set of axioms supports a first quantization net obtained by generalizing the BGL construction. The construction is possible for a large family of Lie groups and provides several new models. We further comment on orthogonal wedges and extension of symmetries.


Introduction
Quantum Field Theory (QFT) lives in a tension between the locality principle and the underlying group of symmetries characterizing the theory. On one hand, it is a physical principle that every interesting quantity of a theory should be deducible by local measurements, namely-in the language of Algebraic Quantum Field Theory (AQFT)-by the structure of the local algebras (see e.g. [Ha96]). On the other hand, the symmetries of a theory provide a feature to describe physical objects, a "key to nature's secrets," as it happens in the standard model [We05,We11].
In AQFT, models are specified by a net of von Neumann algebras associated to causally complete spacetime regions satisfying fundamental quantum and relativistic principles, such as isotony, locality, covariance, positivity of the energy, and existence of a vacuum state. An important bridge between the geometry and the algebraic structure is the Bisognano-Wichmann (BW) property of (A)QFT claiming that the modular group of the algebra associated to any Rindler wedge W inside Minkowski spacetime with respect to the vacuum state implements unitarily the covariant oneparameter group of boosts fixing the wedge W . As a consequence, the algebraic structure of the model, through the Tomita-Takesaki theory, contains the information about the symmetry group acting on the model. Starting with the BW property, one can enlarge the symmetry group of a QFT [GLW98,MT18], find new relations among field theories [GLW98, LMPR19,MR20], establish proper relations among spin and statistics [GL95], and compute entropy in QFT [LX18,Wi18]. For recent results on this property we refer to [Gu19,DM20].
Particles are field-derived concepts that can be described as unitary positive energy representations of the symmetry group. They are building blocks to construct Quantum Field Theories. The operator-valued distribution Φ U defining the free field associated to any particle U is not provided by a canonical construction, see e.g. [BGL02,LMR16]. On the other hand, the von Neumann algebra net generated by Φ U satisfies the Bisognano-Wichmann property and the PCT Theorem 1 . These properties provide the tools for a canonical construction of the free algebra net [BGL02]: Segal's second quantization gives the vacuum representation of the Weyl algebra on the Fock space associated with the one-particle Hilbert space. The Araki lattice of von Neumann algebras is uniquely determined by the local one-particle structure encoded in the lattice of closed real subspaces, the first quantization [Ar63]. As a result of the Tomita-Takesaki modular theory for real subspaces, the set of real states for a particle U localized in a wedge region is uniquely determined by the couple (e −2πKW , U (j W )) where U (j W ) is the antiunitary implementation of the wedge reflection and K W is the generator of the one-parameter group of boosts associated to the wedge W . They satisfy the Tomita relation U (j W )e 2πKW U (j W ) = e −2πKW . The one-particle states and the local algebra associated to bounded causally complete regions are obtained by wedge state spaces and algebra • Locality. Complementary wedges correspond to inverted one-parameter groups of boosts. For instance dilations associated to causally complementary intervals in chiral theory or boosts associated to complementary wedges are inverse to each other. On the abstract wedge space this is captured by defining the complementary wedge of W = (x, σ) by W ′ = (−x, σ).
• Isotony. By the existence of a (positive) invariant cone C in the Lie algebra g, it is possible to define a wedge endomorphism semigroup defining the wedge inclusion relation. Given an Euler wedge W = (x, σ), the generators in the positive cone lying in the subspaces g ±1 define proper wedge inclusions as each of them generates with x a translation-dilation group (isomorphic to the affine group of the real line); see [Bo92,Wi92,Wi93] and in particular [Bo00]. This is the case of wedge endomorphisms in Minkowski spacetime given by lightlike shifting or Möbius transformations mapping an interval into itself as the translations do for the half-lines. These properties define a local partially ordered set of wedges that can support key features of an AQFT structure.
It is important to note that the wedge space only depends on the Lie group and its Lie algebra, and the order structure on the invariant cone C ⊆ g. The relations among the wedges specify the abstract spacetime structure to a large extent. For example, PSL 2 (R) is the symmetry group for the 2-dimensional de Sitter spacetime and for the chiral circle. If one considers PSL 2 (R) with the trivial cone in sl 2 (R)-no proper inclusions of wedges-then it describes a QFT on de Sitter spacetime; if one considers C ⊂ sl 2 (R) as in (2.17), inclusion relations among wedges arise, and we obtain the wedge space on S 1 . 2 This correspondence between isotony and and positivity of the energy was also studied in [GL03]; see also [Bo92,Bo00,Wi92,Wi93] and [NÓ17,Ne19,Ne19b]. For recent classification results for the triples (g, x, C), we refer to [Oeh20,Oeh20b].
There is more interesting structure on the abstract wedge space: • Orthogonal wedges: We call two abstract wedges W 1 = (x 1 , σ 1 ) and W 2 = (x 2 , σ 2 ) orthogonal if σ 1 (x 2 ) = −x 2 , i.e., W 2 is reflected into its complement W ′ 2 . Examples of orthogonal wedges are coordinate wedges on Minkowski spacetime 3 , or the upper and the right half-circle in chiral theories on S 1 . This notion, which immediately generalizes to the abstract setting, plays a central role in spin-statistics relations [GL95] and the nuclearity property in conformal field theory [BDL07].
• Symmetric wedges. A wedge W is called symmetric if there exists g ∈ G ↑ , such that g.W = W ′ . For instance, any couple of wedge regions, in 1 + s-dimensional Minkowski spacetime with s ≥ 2, are transformed one into the other by the action of the Poincaré group G ↑ = P ↑ + . On the other hand, in 1 + 1-dimensional Minkowski space, the right and the left wedges are not symmetric. Indeed W R = {(t, x) ∈ R 1+1 : |t| < x} and W L = {(t, x) ∈ R 1+1 : |t| < −x} belong to disjoint transitive families with respect to the P ↑ + -action. Further examples of symmetric wedges are intervals in conformal theories on the circle. Half-lines in the real line are not symmetric wedges with respect to the translation-dilation group. A transitive family of wedges has the feature that algebras associated to complementary wedges are -by covariance -unitary equivalent. On the other side, there is no contradiction in having a G ↑ -covariant net of von Neumann algebras on a transitive family of non-symmetric wedges with trivial algebras associated to the family of complements.
In the first part of the paper we define and investigate the abstract structure we have described. When the center Z(G ↑ ) is non trivial, for instances when covering groups are considered, a generalized notion of complementary wedges has to be introduced. Indeed, while Euler elements are uniquely determined as generators of one-parameter groups in G ↑ , several involutions σ satisfying Ad(σ) = e πi ad x can be associated to the same Euler element x. In an analogous way, different wedge complements can be labeled by central elements. We classify wedge orbits and define a notion of a central wedge complement. Furthermore, if W ′ does not belong to the G ↑ -orbit of W , a new action of G on the wedge space is defined. This happens for instance in fermionic nets.
Having specified the abstract structures, we are prepared to answer the following question: "Which Lie algebras/groups support such a structure?" To this end, we first classify Euler elements in real simple Lie algebras in Theorem 3.10. The key point of this classification is that Euler elements are conjugate under inner automorphisms to elements in any given Cartan subspace of hyperbolic elements. Here the restriction to simple Lie algebras is not restrictive because any symmetric Euler element is contained in a semi-simple Lie subalgebra. Furthermore, an Euler element is symmetric if and only if it is contained in an sl 2 (R)-subalgebra (see Theorem 3.13 for these results). As a consequence, there is a large family of real Lie algebras supporting such wedge structures which properly contains the well known models. Note that, for a Lie algebra g containing an Euler element x ∈ g, there always exists a graded Lie group G with Lie algebra g and a corresponding Euler wedge (x, σ).
The second part of the paper is devoted to nets of standard subspaces. Is it possible to construct one-particle models supporting this abstract setting? Starting with a G ↑orbit W + in the wedge space, we describe a set of axioms which, for the well known models, reflect fundamental quantum and relativistic principles corresponding to the one-particle Haag-Kastler axioms. This set of axioms is fulfilled by extending the BGL construction to every graded Lie group G, supporting a suitable wedge space. A twisted locality relation among complementary wedges is introduced in order to relate central complementary wedges.
Do we get any new models out of this general construction? The answer is affirmative. All the simple Lie algebras whose restricted root system appears in Theorem 3.10 correspond to a graded Lie group with a non-trivial wedge space. There are for instance Lie algebras of type E 7 that do not correspond to any known models. In this context the Jordan spacetimes of Günaydin [Gu93,Gu00,Gu01] and the simple spacetime manifolds in the sense of Mack-de Riese [MdR07] are homogeneous spaces of simple hermitian Lie groups whose Lie algebras contain Euler elements, and the corresponding abstract wedges correspond to domains in these causal manifolds. These Lie groups have many (anti-)unitary representations, some of them with positive energy with respect to a non-trivial invariant cone C in the Lie algebra. As a consequence, they support many one-particle nets [NÓ20] and second quantization models of von Neumann algebras whose physical meaning has to be investigated.
The structure of this paper is as follows: In Section 2 the wedge space is defined and its properties are studied. A number of examples are discussed in detail to show how the abstract setting applies to the known models and realizes the well known structure. In Section 3 we study the Euler elements in Lie algebras. We relate orthogonal and symmetric wedges and provide a classification of Lie algebras supporting (symmetric) Euler elements. In Section 4 we apply this structure to define and construct one-particle nets associated to graded Lie groups supporting a wedge structure. We further stress new models, orthogonal wedges and extension of symmetries. We hope this paper is approachable for the Lie Theory community as well as the Algebraic Quantum Field Theory community.

The abstract setting
In this section we develop an abstract perspective on wedge domains in spacetimes, phrased completely in group theoretic terms. As wedge domains are supposed to correspond to standard subspaces in Hilbert spaces, we orient our approach on how standard subspaces are parametrized.
Let Stand(H) denote the set of standard subspaces of the complex Hilbert space H. In Section 4 we shall see that every standard subspace V determines a pair (∆ V , J V ) of modular objects and that V can be recovered from this pair by V = Fix(J V ∆ 1/2 V ). This observation can be used to obtain a representation theoretic parametrization of Stand(H): each standard subspace V specifies a continuous homomorphism (2.1) We thus obtain a bijection between Stand(H) and the set Hom gr (R × , AU(H)) of continuous morphisms of graded topological groups. The space Stand(H) carries three important features: • an order structure, defined by set inclusion • the action of AU(H) as a symmetry group.
The order structure is hard to express in terms of the modular groups (see [Ne19b] for some first steps in this direction), but the duality operation corresponds to inversion and the action of AU(H) translates into where ε(g) = 1 if g is unitary and ε(g) = −1 otherwise. So unitary operators g ∈ U(H) simply act by conjugation, but antiunitary operators also involve inversion. In particular, We now develop the corresponding structures by replacing AU(H) by a finite dimensional graded Lie group.

Group theoretical setting
The basic ingredient of our approach is a finite dimensional graded Lie group (G, ε G ), i.e., G is a Lie group and ε G : G → {±1} a continuous homomorphism. We write G is a normal subgroup of index 2 and G ↓ = G \ G ↑ . We also fix a pointed closed convex cone C ⊆ g satisfying Ad(g)C = ε G (g)C for g ∈ G. (2.4) As we shall see in the following, for graded Lie groups, it is more natural to work with the twisted adjoint action Ad ε : G → Aut(g), Ad ε (g) := ε G (g) Ad(g), (2.5) so that (2.4) actually means that C is invariant under the twisted adjoint action. The cone C will play a role in specifying an order structure. It is related to positive spectrum conditions on the level of unitary representations. We also allow C = {0}. For instance, the Lie algebra g = so 1,d (R) of the Lorentz group G = O 1,d (R), the isometry group of de Sitter space time dS d , contains no non-trivial invariant cone.

The space
Hom gr (R × , G) and abstract wedges In this section we define the fundamental objects we will need in the forthcoming discussion. We write Hom gr (R × , G) for the space of continuous morphisms of graded Lie groups R × → G, where R × is endowed with its canonical grading by ε(r) := sgn(r). On this space G acts by where the twist is motivated by formula (2.2). Elements of G ↑ simply act by conjugation.
Since we are dealing with Lie groups, we also have the following simpler description of the space Hom gr (R × , G) by the set Proposition 2.1. The map is a bijection. It is equivariant with respect to the action of G on G by g.(x, σ) := (Ad ε (g)x, gσg −1 ). (2.8) Note that center Z(G ↑ ) of G ↑ acts trivially on the Lie algebra but it may act non-trivially on involutions in G ↓ .
Remark 2.2. For every involution σ ∈ G ↓ , the involutive automorphism σ G (g) := σgσ defines the structure of a symmetric Lie group (G ↑ , σ G ), and G ∼ = G ↑ ⋊ {id, σ}, so that we can translate between G as a graded Lie group and the pair (G ↑ , σ G ), without loosing information.
To indicate the analogy of elements of G with the wedge domains in QFT, we shall often denote the elements of G by W = (x, σ). Definition 2.3. (a) We assign to W = (x, σ) ∈ G the one-parameter group Then we have the graded homomorphism Note that Ψ(γ W ) = W in terms of (2.7).
Definition 2.4. (a) We call an element x of the finite dimensional real Lie algebra g an Euler element if ad x is diagonalizable with Spec(ad x) ⊆ {−1, 0, 1}, so that the eigenspace decomposition with respect to ad x defines a 3-grading of g: (see [BN04] for more details on Euler elements in more general Lie algebras). Then σ x (y j ) = (−1) j y j for y j ∈ g j (x) defines an involutive automorphism of g.
For an Euler element we write O x = Inn(g)x ⊆ g for the orbit of x under the group Inn(g) = e ad g of inner automorphisms. 4 We say that We write E(g) for the set of non-zero Euler elements in g and E sym (g) ⊆ E(g) for the subset of symmetric Euler elements. (b) An element (x, σ) ∈ G is called an Euler couple or Euler wedge if Ad(σ) = e πi ad x . (2.10) Then σ is called an Euler involution. We write G E ⊆ G for the subset of Euler couples and note that the relation e πi ad x = e −πi ad x implies that the subset G E is invariant under the G-action.
For an Euler element x ∈ E(g), the relation (2.10) only determines σ up to an element z ∈ G ↑ ∩ker(Ad) for which (σz) 2 = e, i.e., σzσ = z −1 . Note that, if G ↑ is connected, then G ↑ ∩ker(Ad) = Z(G ↑ ) is the center of G ↑ . The couples (x, σ) that we have seen in the physics literature are all Euler couples (cf. [NÓ17,Ex. 5.15]). This ensures many properties, such as the proper relation between spin and statistics, see for instance [GL95].
• g(W ) := L W − L W , the Lie algebra generated by L W .
• the semigroup associated to the triple (C, x, σ): As the unit group of S W is given by , the semigroup S W defines a G ↑ -invariant partial order on the orbit G ↑ .W ⊆ G by (2.11) In particular, g.W ≤ W is equivalent to g ∈ S W .
We have the following relations among these objects: Lemma 2.6. For every W = (x W , σ W ) ∈ G, g ∈ G, and t ∈ R, the following assertions hold: Proof. (i) For W = (x, σ) ∈ G, the first two relations follow from the fact that exp(Rx) commutes with x and σ. The second follows from (v) The assertion is clear for g ∈ G ↑ . For g ∈ G ↓ , we have gσ ∈ G ↑ , so that This implies in particular that L g.W = Ad(g)L W . From G ↑ g.W = gG ↑ W g −1 , we thus obtain S g.W = gS W g −1 .
In this discussion we started with a Lie group. We remark that one can also start with a Lie algebra as follows: Consider a quadruple (g, σ g , h, C) of a Lie algebra g, an involutive automorphism σ g of g, and a pointed closed convex invariant cone C ⊆ g with σ g (C) = −C. Then σ g integrates to an automorphism σ G of the 1-connected Lie group G ↑ with Lie algebra g, so that we obtain all the data required above with G : For two such quadruples (g j , τ g,j , h j , C j ) j=1,2 , a homomorphism ϕ : g 1 → g 2 of Lie algebras is compatible with this structure if ϕ • τ g,1 = τ g,2 • ϕ, ϕ(h 1 ) = h 2 and ϕ(C 1 ) ⊆ C 2 .
We thus obtain a category whose objects are the quadruples (g, τ g , h, C) and its morphisms are the compatible homomorphisms. A similar category can be defined on the group level, but there are some subtle ambiguities concerning the possible extensions of the group structure from G ↑ to G.
Remark 2.7. (Twisted extensions of G ↑ to G) We start with a graded group G for which G ↓ contains an involution σ, so that G ∼ = G ↓ ⋊ {e, σ}, where σ acts on G ↑ by the automorphism σ G (g) := σgσ. This defines a split group extension and we are now asking for other group extensions for which the elements in G ↓ define the same element in the group Out(G ↑ ) = Aut(G ↑ )/ Inn(G ↑ ) of outer automorphisms of G ↑ . These extensions are parametrized by the group by assigning to z ∈ Z(G ↑ ) + the group structure on G ↑ × {1, −1} given by (g, 1)(g ′ , ε ′ ) = (gg ′ , ε ′ ), (e, −1)(g ′ , 1) = (σ G (g ′ ), −1) and (e, −1) 2 = (z, 1). (2.12) We write G z for the corresponding Lie group. Basically, this means that the element σ := (e, −1) has the same commutation relations with G ↑ but its square is z instead of e: σg σ −1 = σ G (g) for g ∈ G and σ 2 = z.
(e) As we shall see in Example 2.13 below, it may happen that, for the twisted groups G z , the coset G ↓ z contains no involutions. In this example G ↑ = SL 2 (R) and G = G ↑ {e, γ} with γ 2 = −1.
(2.15) Hence G ↓ z contains an involution if and only if If z = gσ G (g) for some g ∈ G ↑ , then conjugating with g implies that g and σ G (g) commute.
The discussion in Example 2.13 shows that (2.15) is not satisfied for z = −1 and the Euler involution of G ↑ = SL 2 (R). For any odd degree covering SL 2 (R) (2k+1) → SL 2 (R), the central involution is mapped onto −1, so that this observation carries over to odd coverings of SL 2 (R).
The situation changes if we consider G ↑ = SL 2 (C) instead. Then g := i 1 0 0 −1 satisfies g 2 = −1, so that the group G = G ↑ {1, σ} with σ 2 = −1 contains the non-trivial involution g σ ∈ G ↓ . As this involution is central, G ∼ = G ↑ × Z 2 is a direct product. . This is equivalent to the existence of an element g ∈ G ↑ with g.W 0 = W ′ 0 , i.e., g ∈ (G ↑ ) τ with Ad(g)h = −h. (b) If W 0 is an Euler couple, then W(W 0 ) is a family of Euler couples, and we shall see below that in this case we have W(W 0 ) = W + (W 0 ) in many important cases.

The abstract wedge space, some fundamental examples
We collect some fundamental examples, starting from the low dimensional cases, that we shall refer to throughout the paper.
The cone C = R + × {0} ⊆ g satisfies the invariance condition (2.4) and the corresponding semigroup S W is Therefore the map W + (W ) ∋ g.(λ, r 0 ) → gW defines an order preserving bijection between the abstract wedge space W + (W ) ⊆ G and the set I + (R) = {(t, ∞) : t ∈ R} of of lower bounded open intervals in R. Accordingly, we may write W (t,∞) = (Λ (t,∞) , r t ) := ζ(t)W = (Ad(ζ(t))λ, r t ) for t ∈ R. Acting with reflections, we also obtain the couples corresponding to past pointing half-lines (−∞, t) ⊂ R. We thus obtain a bijection between the full wedge space W(W ) and the set I(R) of open semibounded intervals in R. We shall denote with δ I the one-parameter group of dilations with generator λ I corresponding to the half line I.
The set E(g) = Ad(G ↑ ){±λ} of non-zero Euler elements in g consists of two G ↑ -orbits and, for each non-zero Euler element ± Ad(ζ(t))λ ∈ E(g), the reflection r t is the unique partner for which (± Ad(ζ(t))λ, r t ) ∈ G. Accordingly, Euler couples in G are in one-to-one correspondence with semi-infinite open intervals in R.
On the compactified line, the point reflection τ (x) = −x in 0 acts on the Lie algebra by (2.16) The infinitesimal generator h := of δ is an Euler element and W := (h, τ ) is an Euler couple. Since Möb 2 ∼ = PGL 2 (R) ∼ = Aut(sl 2 (R)), for any Euler couple (x, τ ), the involution τ is determined by the requirement that it acts on g = sl 2 (R) by e πi ad x . We conclude that the action of G ↑ = Möb on the set of Euler couples is transitive, i.e., To see the geometric side of Euler couples, let us call a non-dense, non-empty open connected subset I ⊆ S 1 an interval and write I(S 1 ) for the set of intervals in S 1 . It is easy to see that Möb acts transitively on I(S 1 ). To determine the stabilizer of an interval, we consider the upper half circle, which corresponds to the half line (0, ∞) ⊆ R. Each element g ∈ Möb mapping (0, ∞) onto itself fixes 0 and ∞. Since it is completely determined by the image of a third point, it is of the form δ(t) if g.1 = e t . Therefore the stabilizer of (0, ∞) in Möb is the subgroup δ(R), which coincides with the stabilizer of h under the adjoint action. This already shows that W + (W ) and I(S 1 ) are isomorphic homogeneous spaces of Möb. In particular, we can associate to an interval I = g(0, ∞) the reflection τ I = gτ g −1 and the one-parameter group δ I := gδg −1 . Note that τ I is an orientation reversing involution mapping I to the complementary open interval I ′ . We write x I := Ad(g)h for the infinitesimal generator of δ I , so that the assignment I → x I defines an equivariant bijection I(S 1 ) → E(g). The anticlockwise orientation of S 1 , which can also be considered as a causal structure, is used here to pick the sign of x I in such a way that the flow δ I is counter clockwise (future pointing) on I. Accordingly, x I ′ = −x I corresponds to the complementary interval I ′ .
To identify the natural order on the abstract wedge space G E = W + (W ), we consider for This shows that is a pointed generating invariant cone in g. The Lie wedge specified by the triple (h, τ, C) is We further have G(W ) = G ↑ , and the associated semigroup is Therefore the map defines an order preserving bijection between the abstract wedge space W(W ) and the ordered set I(S 1 ).
(d) We now consider the universal covering of the Möbius group Möb. Concretely, we put G := Möb ⋊ {1, τ }, where τ acts on Möb by integrating Ad(τ ) from (2.16) to an automorphism of Möb. The group G is a graded Lie group and G ↑ := Möb is its identity component. We have a covering homomorphism q G : G → Möb 2 whose kernel Z( Möb) ∼ = Z is discrete cyclic. We write ρ, δ, ζ and ζ ∪ for the canonical lifts of the one-parameter groups ρ, δ, ζ, ζ ∪ of Möb, P + := δ(R) ζ(R), The action of Möb on S 1 lifts canonically to an action of the connected group G ↑ = Möb on the universal covering S 1 ∼ = R, where we fix the covering map q S 1 : R → R, defined by q S 1 (θ) = ρ(θ).0, which corresponds to the map θ → e iθ = C( ρ(θ).0) in the circle picture. We may thus identify S 1 with the homogeneous space Möb/ P − ∼ = R. As conjugation with τ on Möb preserves the subgroup P − , it also acts on S 1 . From (2.16) it follows that it simply acts by the point reflection τ .x = −x in the base point 0. We also note that Z := ker(q G ) = ρ(2πZ) is the group of deck transformations of the covering q S 1 , which acts by ρ(2πn).x = x + 2πn for n ∈ Z. (2.19) We call a non-empty interval I ⊆ R admissible if its length is strictly smaller than 2π and write I(R) for the set of admissible intervals. An interval I ⊆ R is admissible if and only if there exists an interval I ∈ I(S 1 ) such that I is a connected component of q −1 S 1 (I). The group Z acts transitively on the set of these connected components. As Möb acts transitively on I(S 1 ), it follows that the group Möb acts transitively on the set I(R), and that composition with q S 1 yields an equivariant covering map (2.20) We further have: • The group P + = δ(R) ζ(R) fixes the points {(2k + 1)π : k ∈ Z}.
• For I ∈ I(S 1 ), let δ I be the lift of the one-parameter group δ I . Then δ I preserves every interval in the preimage q −1 S 1 (I).
• The inverse images of τ ∈ Möb 2 in Möb 2 are the elements τ n := ρ(2πn) τ , n ∈ Z. These are involutions, acting by which is a point reflection in the point πn. All pairs (h, τ n ) are Euler couples in G( Möb 2 ), and from the discussion of the set of Euler couples G E (Möb 2 ) under (c), we know that the involutions τ n exhaust all possibilities for supplementing h to an Euler couple.
There is an interesting difference to the situation for Möb 2 , where Möb acts transitively on the set G E (Möb 2 ) of Euler couples. To see what happens for Möb 2 , recall that the stabilizer of the element (h, τ ) ∈ G E (Möb 2 ) in Möb is the subgroup δ(R). Its inverse image is the group An element g ∈ Möb fixes (h, τ n ) if and only if Ad(g)h = h and g τ n g −1 = τ n . The first condition is equivalent to g being of the form The second condition is equivalent to τ g τ = τ n g τ n = g, which takes the form and this is equivalent to k = 0. We conclude that the stabilizer of (h, τ n ) is We also note that We conclude that the group Möb does not act transitively on the set G E of Euler couples. It has two orbits: (2.23) We also refer to Example 2.14 for a discussion of this issue from a different perspective.
• The subgroup δ(R) preserves every interval which is a non-trivial orbit of δ(R), acting on R.
If, conversely, g ∈ Möb preserves such an interval, then its image in Möb is contained in δ(R), As every open orbit of δ(R) is an interval of length π, the element g can only preserve such an orbit if k = 0. This shows that Möb (h, τn) also is the stabilizer group of any open δ(R)-orbit in R. We conclude that, for the Euler couple W 0 = (h, τ 0 ), the map defines a G ↑ -equivariant bijection between the abstract wedge space W + (W 0 ) ⊆ G and the set I(R) of admissible intervals in R. Since the full group G acts on the space I(R) of intervals, Φ can be used to transport this action to a G-action on the space W + (W 0 ), extending the action of the subgroup G ↑ . Since τ 0 (0, π) = (−π, 0) = ρ(−π)(0, π), we have Here we use that ρ(−2π) ∈ Z(G ↑ ). Note that we have chosen (0, π) to be the image of W 0 throught Φ. Further possible actions come from the identifications and one can likewise see that g * αn (x, σ) := (Ad ε (g), α n gσg −1 ) for g ∈ G ↓ and α n = ρ((2n − 1)2π) ∈ Z(G ↑ ), extends the action of G ↑ on W + (W n ) to G and Φ = Φ 0 for n = 0 (see also (2.37) and Section 2.4.2 for this kind of action).
(e) Let q : Möb (n) → Möb be the n-fold covering group of Möb and ρ (n) , δ (n) , ζ (n) and ζ (n) ∪ be the lifts of the corresponding one-parameter groups of Möb. We further put P −,(n) := δ (n) (R)ζ (n) ∪ (R), so that we obtain an n-fold covering of the circle, and the action of the one-parameter group ρ (n) induces a diffeomorphism The set of wedges can be described analogously to the case (d), but there is a difference depending on the parity of n. If n is even, the group G ↑ has two orbits in the set G E of Euler couples, but if n is odd, there is only one. Indeed, for n = 2k, the element ρ (n) (2πk) acts as an involution on S 1 n . So it fixes all Euler couples (h, τ n ), even if it does NOT fix any proper interval in S 1 n (see also Example 2.14).
(f) The example arising most prominently in physics is the proper Poincaré group It acts on 1 + d-dimensional Minkowski space R 1,d as an isometry group of the Lorentzian metric given by ( for the open future light cone, the grading on G is specified by time reversal, i.e., gV + = ε(x, g)V + . In particular C := V + is a pointed closed convex cone satisfying (2.4). For d > 1, this is, up to sign, the only non-zero pointed invariant cone in the Lie algebra g.
The generator k 1 ∈ so 1,d (R) of the Lorentz boost on the (x 0 , x 1 )-plane is an Euler element. It combines with the spacetime reflection j 1 (x) = (−x 0 , −x 1 , x 2 , . . . , x d ) to the Euler couple (k 1 , j 1 ). We associate to (k 1 , j 1 ) the spacetime region the standard right wedge, and note that W 1 is invariant under exp(Rk 1 ). It turns out that the semigroup S (k1,j1) associated to the couple (k 1 , j 1 ) in Definition 2.5 satisfies [NÓ17,Lemma 4.12]). From (2.27) it follows that the map defines an order preserving bijection between the abstract wedge space W ⊆ G and the set of wedge domains in Minkowski space R 1+d . For an abstract wedge W = (k W , j W ) ∈ W, the Euler element k W is the corresponding boost generator. For an axial wedge W i := {x ∈ R 1+d : |x 0 | < x i }, i = 1, . . . , n, the corresponding Euler couple will be denoted (k i , j i ).

Nets of wedges, isotony, central locality and covering groups
In the following sections we will focus on the description of relative positions of wedges, in particular wedge inclusions and the locality principle.

Wedge inclusion
Firstly consider this wedge inclusion configuration called half-sided modular inclusion: We then call W 1 ≤ W 0 a ±half-sided modular inclusion.
The next lemma shows that any wedge inclusion can be described in terms of positive and negative half-sided modular inclusions.

Central locality
For a wedge W = (x, σ), the dual wedge W ′ = (−x, σ) need not be contained in the orbit W + = G ↑ .W . If, however, G ↑ has a non-trivial central subgroup Z such that, modulo Z, the complement W ′ is contained in W + , then we use central elements α ∈ Z to define "twisted complements" W ′ α which are contained in W + , and this in turn leads to a twisted action of the full group G on W + . We also obtain on W + a complementation map W → W ′ α .
Let Z ⊆ Z(G ↑ ) be a closed normal subgroup of G, and q : G → G := G/Z be the corresponding surjective morphism of graded Lie groups with kernel Z. If Z is discrete, then q is a covering map. The morphism of graded Lie groups q induces a natural map where Ad g : G → Aut(g) denotes the factorized adjoint action which exists because Z = ker(q) acts trivially on g. It restricts to a map As the following example shows, neither of these maps is always surjective. The main obstruction is that, although the differential L(q) : L(G) → L(G) is surjective, there may be involutions τ ∈ G ↓ for which no involution σ ∈ G ↓ with q(σ) = τ exists. This phenomenon is tightly related to the twisted groups G z discussed in Remark 2.7 because these twists disappear for z ∈ Z in G/Z ∼ = G/Z. Example 2.13. We consider the graded Lie group It has two connected component and G ↑ = SL 2 (R). 6 The subgroup Z := {±1} is central and the quotient map q : G → G := G/Z is a 2-fold covering. The Euler element x := 1 2 1 0 0 −1 ∈ g = sl 2 (R) combines with the involution q(γ) ∈ G ↓ to the Euler couple (x, q(γ)) ∈ G. However, the set This is equivalent to a = −d and b = c = 0, contradicting that 1 = det(g) = −a 2 . We conclude in particular that the maps G(G) → G and G E (G) → G E (G) are not surjective.
We now discuss G ↑ -orbits in G(G).
In the examples we have in mind, the central subgroup Z is discrete. Involution lifts and central wedge orbit. Each element σ ∈ G ↓ acts in the same way on the abelian normal subgroup Z by the involution which restricts to an involution σ Z ∈ Aut(Z) because Z is central in G ↑ and a normal subgroup of G. In the following we shall need the subgroups (2.31) For γ ∈ Z − , the element γ 2 = (γ σ γ −1 ) −1 is contained in Z 1 , so that the quotient group Z − /Z 1 is an elementary abelian 2-group, i.e., isomorphic to Z (B) 2 for some index set B. For an involution σ ∈ G ↓ and β ∈ Z(G ↑ ), the element βσ ∈ G ↓ is an involution if and only if β ∈ Z − . Therefore α * (x, σ) := (x, ασ) (2.32) defines an action of Z − on G(G), commuting with the conjugation action of G ↑ and satisfying , the fiber over W := (x, q(σ)) is thus given by The subgroup Z ⊆ G ↑ acts by conjugation on the fiber Z − * W : so that the quotient group Z − /Z 1 parametrizes the Z-conjugation orbits in the fiber Z − * W . 7 Here is an example.
Example 2.14. (a) If Z ∼ = Z and n σ = −n, then Z − = Z and Z 1 = 2Z, so that Z − /Z 1 ∼ = Z/2Z. (b) If Z = Z n and n σ = −n, then Z − = Z n and Z 1 = 2Z n , so that In general the group G ↑ does not act transitively on the inverse image of the orbit W + := G ↑ .W ⊆ G under q G . We now describe how this set decomposes into orbits. By the transitivity of the G ↑ -action on W + , it suffices to consider the orbits of the stabilizer Example 2.10(d) and the canonical homomorphism is an Euler couple mapped to W = (h, σ) ∈ G. As z τ = z −1 for every z ∈ Z, we have Z = Z − and Z 1 = 2Z is a subgroup of index 2. To calculate Z 2 , we observe that We conclude that The following definition generalizes the notion of complementary wedge given in Definition 2.5 (a).
We will refer to couples of the form W ′ α as complementary wedges. We consider W ′ α as a "complement" of W because q G maps W ′ α to W ′ (see item (a) below).
Example 2.19. We show that for G = Möb ⋊ {1, τ} as in Example 2.10(d), we have to use twisted complements to obtain a G ↑ -orbit in G E (G) invariant under complementation. We have already seen that G E ( Möb) contains two G ↑ -orbits, represented by the couples W 0 = (h, τ ) and W 1 = (h, τ 1 ).
so that complementation exchanges the two G ↑ -orbits in G E ( Möb). On the other hand, for the action * α defined in (2.37), the full group G preserves both G ↑ -orbits. Since Ad(ρ(−π))h = −h, the element g := ρ(−π) can be used to define a suitable α-twisted conjugation as follows. We note that Thus G E ( Möb 2 ) consists of two G ↑ -orbits, none of which is invariant under complementation, but both are invariant under α-complementation. An analogous computation leads to the same picture for even coverings of Möb, in particular for the fermionic case.

Euler elements and 3-graded Lie algebras
In this section we exhibit a general relation between two notions that are a priori unrelated: complementary and orthogonal wedges. For the sake of simplicity we consider in this introductory part the case of the Poincaré group G = P + on R 1+2 (cf. Example 2.10). We have seen that if W = (k W , j W ) is a wedge of the group G, then W ′ = (−k W , j W ) is the opposite wedge. The π-spatial rotation ρ(π) takes W onto W ′ and vice versa. Thus there exists a group element g ∈ G ↑ = P ↑ + such that Ad(g)k W = −k W , and in this sense k W is symmetric. This ensures a symmetry between a wedge and its opposite wedge, which corresponds to its causal complement in Minkowski spacetime.
Typical pairs of orthogonal wedges are the coordinate wedges The importance of this couple of wedges comes by the clear geometric relation: the wedge reflection of W 1 acts on the orthogonal wedge as In [GL95] the authors study the orthogonality relation in order to extend the unitary covariance representation of the Poincaré group P ↑ + to an (anti-)unitary representation of the graded group P + and establish the Spin-Statistics Theorem. In this extension process, orthogonal Euler wedges play a crucial role. This point will be discussed from our abstract perspective in Section 4.4 below.
In this section we will see how, in our setting, the existence of a symmetric Euler element in the Lie algebra ensures the existence of an orthogonal pair. For symmetric Euler elements, the orthogonality relation for Euler elements is symmetric, and orthogonal pairs of Euler elements generate a subalgebra isomorphic to sl 2 (R) in g.

Preliminaries on Lie algebras and algebraic groups
In this subsection we collect some basic facts on finite dimensional real Lie algebras and on real algebraic groups (see [HN12] for Lie algebras and [Ho81] for algebraic groups).
A Lie algebra g is called simple if g and {0} are the only ideals of g. It is called semisimple if it is a direct sum of simple ideals g = g 1 ⊕ · · · ⊕ g n . On the other side of the spectrum, we have solvable Lie algebras. These are the ones for which the derived series defined by D 0 (g) := g and D n+1 (g) := [D n (g), D n (g)] satisfies D N (g) = {0} for some N ∈ N. Here [g, g] = span{[x, y] : x, y ∈ g} is the commutator algebra of g.
The fundamental theorem on the Levi decomposition asserts that, if r is the maximal solvable ideal of g, then there exists a semisimple subalgebra s (a Levi complement), such that g ∼ = r ⋊ s is a semidirect sum, i.e., a vector space direct sum of the ideal r and the subalgebra s.
A key feature in the structure theory of semisimple real Lie algebras is the concept of a compactly embedded subalgebra. A subalgebra k ⊆ g is said to be compactly embedded if the subgroup Inn g (k) = e ad k ⊆ Aut(g) has compact closure. We write Inn(g) := Inn g (g) for the subgroup of inner automorphisms of g.
An element x ∈ g is called • elliptic, if ad x is semisimple with purely imaginary eigenvalues, which is equivalent to the one-dimensional Lie subalgebra Rx being compactly embedded.
• hyperbolic, if ad x is diagonalizable.
The Cartan-Killing form κ : g × g → R, κ(x, y) := tr(ad x ad y) is a symmetric bilinear form on g invariant under the automorphism group Aut(g). Recall that a finite dimensional real Lie algebra is semisimple if and only if κ is non-degenerate (Cartan's criterion). Note that κ(x, x) = tr((ad x) 2 ) ≥ 0 if x is hyperbolic and κ(x, x) ≤ 0 if x is elliptic.
In the proof of Proposition 3.2 below we shall use some results from the theory of linear algebraic groups. We now recall the basic concepts. If V is a finite dimensional real vector space, then GL(V ) denotes the group of linear automorphisms of V . Any polynomial function on the linear space End(V ) defines a function on the group GL(V ) and we call a subgroup G ⊆ GL(V ) algebraic if it is the zero set of a family of polynomial functions p j : End(V ) → R. An algebraic group G is said to be • unipotent, if there exists a flag of linear subspaces such that (g − 1)F j ⊆ F j−1 for j = 1, . . . , n and g ∈ G.
In this context one has a decomposition theorem (the Levi decomposition), asserting that every algebraic subgroup G ⊆ GL(V ) is a semidirect product G ∼ = U ⋊ L, where U is unipotent and L is reductive. Moreover, for every reductive subgroup L 1 ⊆ G there exists an element g ∈ G with gL 1 g −1 ⊆ L ([Ho81, Thm. VIII.4.3]).

Symmetric and orthogonal Euler elements
Proposition 3.2. The following assertions hold: (i) An Euler element h ∈ g is symmetric, i.e., −h ∈ O h , if and only if h is contained in a Levi complement s and h is a symmetric Euler element in s.
(ii) If g = r ⋊ s is a Levi decomposition. imply that g = r 0 (h) + [g, g]. As [g, g] is an ideal and r 0 (h) a subalgebra of g, the subgroup Inn g ([g, g]) of Inn(g) is normal, and Inn(g) = Inn g ([g, g]) Inn(r 0 (h)). As Inn(r 0 (h)) fixes h, this in turn shows that and h s ∈ E(s), we thus find x ∈ [g, r] and s ∈ Inn g (s) such that 8 Applying the Lie algebra homomorphism q to both sides, we derive from q(h z ) = 0 and q • e ad x = q that −h s = s.h s , and therefore e ad x h s = h s + 2h z .
We conclude that the unipotent linear map e ad x preserves the linear subspace Rh s + Rh z , and this implies that ad x = log(e ad x ) also has this property. We thus arrive at so that we must have x ∈ g 0 (h) = g 0 (h s ), which in turn leads to 0 = e ad x h s − h s = 2h z , i.e., h = h s ∈ s.
To prove the second assertion of (i), we observe that the homomorphism q : g → s ∼ = g/r satisfies Hence q(E sym (g)) ⊆ E sym (s). If, conversely, h ∈ E sym (s), then we clearly have −h ∈ Inn g (s)h ⊆ Inn(g)h, so that h ∈ E sym (g).
Proposition 3.2 reduces the description of symmetric Euler elements up to conjugation by inner automorphisms to the case of simple Lie algebras.
Remark 3.3. Suppose that g is a finite dimensional Lie algebra containing a pointed generating invariant cone C. If g is not reductive, then C ∩ z(g) = {0} ([Ne99, Thm. VII.3.10]). If τ = σ h is an involution defined by a symmetric Euler element h, then τ fixes every central element, so that we cannot have τ (C) = −C if g is not reductive.
Examples 3.4. (a) If s is a semisimple Lie algebra and h ∈ s an Euler element, then it also is an Euler element in the semidirect sum T s := |s| ⋊ s, where |s| is the linear subspace underlying s, endowed with the s-module structure defined by the adjoint representation. (b) In the simple Lie algebra g := sl n (R), we write n × n-matrices as block 2 × 2-matrices according to the partition n = k + (n − k). Then is diagonalizable with the two eigenvalue n−k n = 1 − k n and − k n . Therefore h k is an Euler element whose 3-grading is given by g 0 (h) = a 0 0 d : a ∈ gl k (R), d ∈ gl n−k (R), tr(a) + tr(d) = 0 , Example 3.5. For g = sl 2 (R), the Euler element We thus obtain two representatives x ± = ± 1 2 0 1 1 0 of conjugacy classes of orthogonal pairs (h, x) of Euler elements for sl 2 (R). The involution corresponding to x ± is given by which shows in particular that As a consequence of the preceding discussion, we see that the orthogonality relation on E(sl 2 (R)) is symmetric: Lemma 3.6. If (x, y) is an orthogonal pair of Euler elements in sl 2 (R), then σ y (x) = −x, so that (y, x) is also symmetric.
Example 3.7. For g = gl 2 (R), the Euler element and we see, as for sl 2 (R), that the orthogonal Euler elements are given by This shows that σ x± (h) = −h.
(3.4) Therefore gl 2 (R) contains a pair (h, x) of orthogonal Euler elements for which σ x (h) = −h. From h ∈ [g, g] it immediately follows that h is not symmetric. We shall see in Theorem 3.13 below that this pathology of the orthogonality relation on the set of Euler elements does not occur for symmetric Euler elements.
Example 3.8. For g = sl 3 (R), the Euler element where we write matrices as 2 × 2-block matrices according to the partition 3 = 1 + 2. Up to conjugacy under the centralizer of h 1 , the symmetric matrices in Fix(−σ h1 ) are represented by These matrices have three different eigenvalues, so that ad x has five eigenvalues, and thus x cannot be an Euler elements of sl 3 (R). We conclude that, there exists no Euler element x ∈ E(sl 3 (R)) for which (h 1 , x) is orthogonal. We shall see in Theorem 3.13(b) below that this never happens for symmetric Euler elements, but h 1 is not symmetric. It corresponds to h 1 for the root system A 2 in the notation of Section 3.3. Example 3.9. For g = sl 4 (R), the Euler element where we write matrices as 2 × 2-block matrices according to the partition 4 = 1 + 3. Up to conjugacy under the centralizer of h 1 , the symmetric matrices in Fix(−σ h1 ) are represented by They all have three different eigenvalues and ad x has five eigenvalues, so that they are not Euler elements. We conclude that there exists no Euler element x ∈ E(sl 4 (R)) for which (h 1 , x) is orthogonal. This is different for the symmetric Euler element where we write matrices as 2 × 2-block matrices according to the partition 4 = 2 + 2. Up to conjugacy under the centralizer of h 2 , the symmetric matrices in Fix(−σ h2 ) are represented by and, for a = b = 1 2 , these are Euler elements orthogonal to h 2 .

Euler elements in simple real Lie algebras
In this section we take a systematic look at Euler elements in simple real Lie algebras. In particular we determine which of them are symmetric and show that pairs of orthogonal ones generate sl 2subalgebras (Theorem 3.13). For the classification of 3-gradings of simple Lie algebras, we refer to [KA88], the concrete list of the 18 types in [Kan98, p. 600] which is also listed below, and Kaneyuki's lecture notes [Kan00]. Let g is a real semisimple Lie algebra. An involutive automorphism θ ∈ Aut(g) is called a Cartan involution if its eigenspaces k := g θ = {x ∈ g : θ(x) = x} and p := g −θ = {x ∈ g : θ(x) = −x} have the property that they are orthogonal with respect to κ, which is negative definite on k and positive definite on p. Then g = k ⊕ p (3.5) is called a Cartan decomposition. Cartan involutions always exist and two such involutions are conjugate under the group Inn(g) of inner automorphism, so they produce isomorphic decompositions ([HN12, Thm. 13.2.11]). If g = k⊕ p is a Cartan decomposition, then k is a maximal compactly embedded subalgebra of g, x ∈ g is elliptic if and only if its adjoint orbit O x = Inn(g)x intersects k, and x ∈ g is hyperbolic if and only if O x ∩ p = ∅.
For the finer structure theory, and also for classification purposes, one starts with a Cartan involution θ and fixes a maximal abelian subspace a ⊆ p. As a is abelian, ad a is a commuting set of diagonalizable operators, hence simultaneously diagonalizable. For a linear functional 0 = α ∈ a * , the simultaneous eigenspaces is called the set of restricted roots. We pick a set Π := {α 1 , . . . , α n } ⊆ Σ of simple roots. This is a subset with the property that every root α ∈ Σ is a linear combination α = n j=1 n j α j , where the coefficients are either all in Z ≥0 or in Z ≤0 . The convex cone is called the positive (Weyl) chamber corresponding to Π.
We have the root space decomposition g = g 0 ⊕ α∈Σ g α and g 0 = m ⊕ a, where m = g 0 ∩ k.
Now θ(g α ) = g −α , and for a non-zero element x α ∈ g α , the 3-dimensional subspace spanned by x α , θ(x α ) and [x α , θ(x α )] ∈ a is a Lie subalgebra isomorphic to sl 2 (R). In particular, it contains a unique element α ∨ ∈ a with α(α ∨ ) = 2. Then is a reflection, and the subgroup is called the Weyl group. Its action on a provides a good description of the adjoint orbits of hyperbolic elements: Every hyperbolic element in g is conjugate to a unique element in Π ⋆ and, for x ∈ a, the intersection O x ∩ a = Wx is the Weyl group orbit ([KN96, Thm. III.10]). From now on we assume that g is simple. Then Σ is an irreducible root system, hence of one of the following types: A n , B n , C n , D n , E 6 , E 7 , E 8 , F 4 , G 2 or BC n , n ≥ 1 (cf. [Bo90a]). If g is a complex simple Lie algebra, then it is also simple as a real Lie algebra, and a Cartan decomposition takes the form where k ⊆ g is a compact real form. Then a = it, where t ⊆ k is maximal abelian. In particular, the restricted root system Σ(g, a) coincides with the root system of the complex Lie algebra g. This leads to a one-to-one correspondence between isomorphy classes of simple complex Lie algebras and the irreducible reduced root systems. If g is not complex, then neither the isomorphy class of g nor of g C is determined by the root system Σ(g, a). For instance all Lie algebras so 1,n (R) have the restricted root system A 1 with dim a = 1, but their complexifications so n+1 (C) have the root systems B k for n = 2k and D k for n = 2k − 1.
The adjoint orbit of an Euler element in g contains a unique h ∈ Π ⋆ . For any Euler element h ∈ Π ⋆ , we have α(h) ∈ {0, 1} for α ∈ Π because the values of the roots on h are the eigenvalues of ad h. If such an element exists, then the irreducible root system Σ must be reduced. Otherwise, for any root α with 2α ∈ Σ, we must have α(h) = 0 because ad x has only three eigenvalues. As the set of such roots generates the same linear space as Σ, this leads to the contradiction h = 0. This excludes the non-reduced simple root systems of type BC n .
To see how many possibilities we have for Euler elements in a, we recall that Π is a linear basis of a, so that, for each j ∈ {1, . . . , n}, there exists a uniquely determined element of a maximal compactly embedded subalgebra k is non-zero. For hermitian Lie algebras, the restricted root system Σ is either of type C r or BC r (cf. Harish Chandra's Theorem [Ne99, Thm. XII.1.14]), and we say that g is of tube type if the restricted root system is of type C r .
The following theorem lists for each irreducible root system Σ the possible Euler elements in the positive chamber Π ⋆ . Since every adjoint orbit in E(g) has a unique representative in Π ⋆ , this classifies the Inn(g)-orbits in E(g) for any non-compact simple real Lie algebra. For semisimple algebras g = g 1 ⊕ · · · ⊕ g k , an element x = (x 1 , . . . , x n ) is an Euler element if and only if its components x j ∈ g j are Euler elements, and its orbit is Therefore it suffices to consider simple Lie algebras, and for these the root system Σ is irreducible. As every complex simple Lie algebra g is also a real simple Lie algebra, our discussion also covers complex Lie algebras.
Theorem 3.10. Suppose that g is a non-compact simple real Lie algebra, with restricted root system Σ ⊆ a * of type X n . We follow the conventions of the tables in [Bo90a] for the classification of irreducible root systems and the enumeration of the simple roots α 1 , . . . , α n . Then every Euler element h ∈ a on which Π is non-negative is one of h 1 , . . . , h n , and for every irreducible root system, the Euler elements among the h j are the following: Proof. Writing the highest root in Σ with respect to the simple system Π as α max = n j=1 c j α j , we have c j ∈ Z >0 for each j. If h ∈ Π ⋆ is an Euler element, then Π(h) ⊆ {0, 1}, and 1 = α max (h) = n j=1 c j α j (h) implies that at most one value α j (h) can be 1, and then the others are 0, i.e., h = h j for some j ∈ {1, . . . , n}. Moreover, h j is an Euler element if and only if c j = 1. Consulting the tables on the irreducible root systems in [Bo90a], we obtain the Euler elements listed in (3.7).
To determine the symmetric ones, let w 0 ∈ W be the longest element of the Weyl group, which is uniquely determined by w * 0 Π = −Π for the dual action of W on a * . Then h ′ j := w 0 (−h j ) is the Euler element in the positive chamber representing the orbit O −hj . Therefore h j is symmetric if and only if −h j ∈ Wh j , which is equivalent to h ′ j = h j . Using the description of w 0 and the root systems in [Bo90a], now leads to h n−1 for n odd, h n for n even, (3.10) Hence the symmetric Euler elements are given by the list (3.8).
This theorem requires some interpretation. So let us first see what it says about complex simple Lie algebras g. In (3.7) we see that only if g is not of type E 8 , F 4 or G 2 , the Lie algebra g contains an Euler element. As Euler elements correspond to 3-gradings of the root system and these in turn to hermitian real forms g • , where ih j ∈ z(k • ) generates the center of a maximal compactly embedded subalgebra k • ([Ne99, Thm. A.V.1]). We thus obtain the following possibilities. In Table  1, we write g • for the hermitian real form, g for the complex Lie algebra, Σ for its root system, and h j for the corresponding Euler element: Euler element su p,q (C), 1 ≤ p ≤ q BC p (p < q), C p (p = q) sl p+q (C) A p+q−1 h p so 2,2n−1 (R), n > 1 C 2 so 2n+1 (C) B n h 1 sp 2n (R) C n sp 2n (C) C n h n so 2,2n−2 (R), n > 2 C 2 so 2n (C) D n h 1 so * (2n) BC m (n = 2m + 1), C m (n = 2m) so 2n (C) D n h n−1 , h n e 6(−14) BC 2 e 6 E 6 h 1 = h ′ 6 e 7(−25) C 3 e 7 E 7 h 7 , we see that they correspond precisely the 3-gradings specified by symmetric Euler elements, as listed in (3.8). Since the Euler elements h n−1 and h n for the root system of type D n are conjugate under a diagram automorphism, they correspond to isomorphic hermitian real forms. In our context hermitian simple Lie algebras are of particular interest. We therefore collect some of their main properties in the following proposition.
Proposition 3.11. For a simple real Lie algebra, the following assertions hold: (b) Since the restricted root system of a hermitian simple Lie algebra is of type C r or BC r , and the first case characterizes the algebras of tube type, the assertion follows from Theorem 3.10 because C r only permits one class of Euler elements.
There are many types of simple 3-graded Lie algebras that are neither complex nor hermitian of tube type; for instance the Lorentzian algebras so 1,n (R). We refer to [Kan98,p. 600] or [Kan00]. for the list of all 18 types which is reproduced below.
so * (4n) C n h n Herm n (H) 8 so n,n (R) C n h n Alt n (R) 9 e 6 (R) e 6 (C) We conclude this section with some finer results concerning orthogonality and symmetry of Euler elements.
Theorem 3.13. If g is simple and h ∈ E(g), then the following assertions hold: such that (h, x) is orthogonal, then (i) h and x are symmetric, (ii) the Lie algebra generated by h and x is isomorphic to sl 2 (R), and Proof. (a) We split the proof into the two cases, according to whether g is a complex Lie algebra or not. We then reduce the second case to the first one. Case 1: g is complex: A simple complex Lie algebra g contains an Euler element, i.e., it possesses a 3-graded root system, if and only if it has a real form g • which is hermitian, i.e., g = (g • ) C = g • ⊕ig • . This follows for example by comparing the list of irreducible root systems for which Euler elements exist (see (3.7)) with the classification of hermitian simple Lie algebras g • (see [Ne99,Thm. A.V.1] and Table 1). In this case the real Lie algebra g • has a Cartan decomposition g • = k • ⊕ p • and the center z(k • ) is one-dimensional and generated by an element z with Spec(ad z) = {0, ±i} ([Ne99, Thm. A.V.1]). Then h = iz is an Euler element in the complexification g for which k • = ker(ad z)∩g • and [z, g • ] = p • , where ad z| p • is a complex structure on the real vector space p • . The corresponding Euler involution σ h = e πi ad h = e π ad z ∈ Aut C (g) thus restricts to the Cartan involution on g • , corresponding to the decomposition k • ⊕ p • . Accordingly, we obtain A Cartan decomposition of g is obtained by k = k • + ip • and p = p • + ik • . If t ⊆ k • is a maximal abelian Lie subalgebra, then a := it ⊆ p is a maximal abelian subspace which contains h = iz ∈ iz(k • ) ⊆ it. The orthogonality of the pair (h, x) means that x ∈ q = Fix(−σ h ). By [KN96,Cor. III.9], x ∈ E(g) ∩ q is conjugate under the centralizer of h to an element in q ∩ p = p • . Fixing a maximal abelian subspace a • ⊆ p • , we may therefore assume that x is an Euler element for the corresponding restricted root system Σ • := Σ(g • , a • ) ⊆ (a • ) * , which is of type C r or BC r (cf. [Ne99, Thm. XII.1.14]). As we have already observed above, the existence of an Euler element x ∈ a • implies that the restricted root system Σ • is reduced, which excludes the case BC r . Therefore g • is of tube type (cf. Proposition 3.11) and Table 2 thus implies that h is symmetric.
The fact that g • is of tube type implies that x ∈ a • corresponds to the unique Euler element h r for the restricted root system Σ • of type C r (see (3.7)). From (3.8) it now follows that x is symmetric (see also Proposition 3.11). This proves (i).
To verify (ii) and (iii), we observe that the root system C r contains the maximal subset {2ε 1 , . . . , 2ε r } of strongly orthogonal roots, i.e., neither sums nor differences of these roots are roots. The multiplicities of these restricted roots are 1 ([Ne99, Thm. XII.1.14]), and [Ta79,p. 12]). As the roots 2ε j all take the value 1 on the Euler element x ∈ a • , we have x = 1 2 r j=1 (2ε j ) ∨ , which is the diagonal element in sl 2 (R) r , corresponding to is isomorphic to sl 2 (R), the same holds for the real Lie subalgebra of g generated by h and x. Now (ii) and (iii) follow from Lemma 3.6. Case 2: g is not complex: Then g C is a simple complex Lie algebra to which all arguments in Case 1 apply. In particular, the real Lie subalgebra s spanned by h, x and [h, x] is isomorphic to sl 2 (R). This proves (ii) and (iii). As s ⊆ g and all Euler elements in sl 2 (R) are symmetric, we also obtain (i). . From these references we further infer the existence of elements e j ∈ g γj such that, for each j, the subalgebra s j := span R {e j , σ h (e j ), [e j , σ h (e j )]} is isomorphic to sl 2 (R). We normalize e j in such a way that, for x j := [e j , σ h (e j )], we have γ j (x j ) = 1. Then loc. cit. further implies that a q := a ∩ q = span{x j : j = 1, . . . , r} for q := g −σ h is maximal abelian in q p . Since h is a symmetric Euler element and the root system Σ(g, a) is irreducible, h corresponds to some h j in the list (3.8). The restricted root system Σ(g, a q ) is always of type C r . The explicit description of the restricted roots in [Kan98, p. 596] now implies that x := r j=1 x j ∈ a q is an Euler element. By construction, it satisfies σ h (x) = −x, so that (h, x) is orthogonal. This completes the proof.
Corollary 3.14. Let g be a finite dimensional Lie algebra and (h, x) be orthogonal Euler elements such that h is also symmetric. Then the following assertions hold: (a) There exists a Levi complement containing h and x.
(b) The Lie algebra generated by h and x is isomorphic to sl 2 (R).
Proof. By Proposition 3.2(i), there exists a Levi decompositions g = r ⋊ s with h ∈ s. We then have for q := Fix(−σ h ) the decompositions q := g 1 (h) ⊕ g −1 (h) = q r ⊕ q s with q r = q ∩ r and q s = q ∩ s, and x ∈ q is an Euler element, hence in particular hyperbolic. Let a r ⊆ q r be a maximal hyperbolic subspace, i.e., a r is abelian, consists of ad-diagonalizable elements and is maximal with respect to this property. Then a r ⊆ [h, r] ⊆ [g, r] consists also of ad-nilpotent elements, hence is central. As ad h| q is injective, it follows that a r = {0}. By [KN96, Prop. III.5], q s contains a maximal hyperbolic subspace a of q and x is conjugate under Inn g (h) to an element of a ⊆ q s . This proves(a). (b) In view of (a), we may w.l.o.g. assume that g is semisimple, and by Theorem 3.13, which applies to each simple ideal, even that g ∼ = sl 2 (R) r for some r ∈ N. As Aut(sl 2 (R)) ∼ = PGL 2 (R) acts transitively on the set of orthogonal pairs of Euler elements in sl 2 (R) (Example 3.5), we may further assume that h = (h 0 , · · · , h 0 ) and x = (x 0 , · · · , x 0 ) for h 0 = so that the Lie subalgebra generated by x and h is the diagonal in sl 2 (R) r , hence isomorphic to sl 2 (R).
(c) follows directly from (b) and Lemma 3.6.

Covariant nets of real subspaces
In this section we develop an axiomatic setting for covariant nets of standard subspaces parametrized by G ↑ -orbits in G E (G).

Standard subspaces
Here we collect some fundamental notions concerning real subspaces of a complex Hilbert space H with scalar product ·, · , linear in the second argument. We call a closed real subspace , and standard if it is cyclic and separating. The symplectic "complement" of a real subspace H is defined by the symplectic form Im ·, · , namely Let H ⊂ H be a standard subspace, and K ⊂ H be a closed, real linear subspace of H. If ∆ it H K = K for all t ∈ R, then K is a standard subspace of K := K + iK and ∆ H | K is the modular operator of K on K. If, in addition, K is a cyclic subspace of H, then H = K.
The following theorem relates positive generators and inclusions of real subspaces. (4.1) for s, t ∈ R, then the following are equivalent: (1) U (t)H ⊂ H for t ≥ 0; (2) ±P is positive.
Part (a) is also called the One-particle Borchers Theorem. Borchers originally proved it for von Neumann algebras with a cyclic and separating vectors. Part (b) is in [BGL02].
With the notation introduced in Examples 2.10(b), we have seen that any couple (U, H) of a one-parameter group (U t ) t∈R with positive (resp. negative) generator and a standard subspace H satisfying the assumptions of Theorem 4.4(a) defines a unitary, positive energy representation of the affine group Aff(R) ∼ = R ⋊ R × implemented by A representation of Aff(R) can also be obtained by looking at some peculiar relative positions of standard subspaces: The half-sided modular inclusions.

The axiomatics of abstract covariant nets
Hereafter we will make the following assumption on the group G.
Example 4.8. Note that G ↓ may contain involutions which are not Euler. We consider the graded Lie group G := SO 1,n (R) with the identity component G ↑ = SO 1,n (R) ↑ . For n ≥ 2, the Lie algebra g = so 1,n (R) is simple, θ(x) = −x ⊤ is a Cartan involution, and a := so 1,1 (R) ⊆ p (acting on the first two components) is a maximal abelian subspace. As the corresponding restricted root system is of type A 1 , our classification scheme (see (3.8) in Theorem 3.10) implies that all Euler elements in g are conjugate to the one corresponding to the boost generator h(x 0 , . . . , x n ) = (x 1 , x 0 , 0, . . . , 0).
Accordingly, an involution σ ∈ G is Euler if and only if σ or −σ is the orthogonal reflection in a 2-dimensional Lorentzian plane. However, G ↓ contains all reflections of the type In particular neither Fix(τ ) nor Fix(−τ ) must have dimension 2.
We now present the analogs of the one-particle Haag-Kastler axioms and further fundamental properties in our general setting. be a map, also called a net of standard subspaces. In the following we denote this data as (W + , U, N). We consider the following properties: (HK3) Spectral condition: C ⊆ C U := {x ∈ g : − i∂U (x) ≥ 0}. We then say that U is C-positive.
When Z α is trivial, for instance when ∂(G ↑ W ) = {e}, then the central twisted locality reduces to the more familiar locality relation.
Concerning (HK3), note that C U is pointed if and only if ker(U ) is discrete. Therefore the assumption that C is pointed is compatible with the possible existence of representations with discrete kernel satisfying (HK3). Furthermore, if C = {0}, then (HK3) trivially holds.
The following property will be central in our discussion because it connects the modular groups of standard subspaces to the unitary representation U of G ↑ .
If the representation U extends antiunitarily to G we can further require: (HK7) G-covariance: For any α ∈ Z(G ↑ ) − such that W ′ α ∈ W + , there exists an (anti-)unitary extension U α of U from G ↑ to G such that the following condition is satisfied: where * α is the α-twisted action (2.37) of G on W + defined in Lemma 2.18(e).
It is enough to provide an extension U α w.r.t. one α ∈ Z(G ↑ ) − such that W ′ α ∈ W + . All the other extensions come as described in Lemma 2.18(f). The modular conjugation of standard subspaces can have a geometric meaning when the extension U α from (HK7) has the following specific form: (HK8) Modular reflection: U α (σ W ) = Z α J N(W ) for α ∈ Z(G ↑ ) − , W ∈ W + with W ′ α ∈ W + and Z α as in (4.3).
In the next sections we will show that there exist nets of standard subspaces satisfying all the above assumptions. It is the analog of the BGL construction in this general setting.

Wedge isotony and half-sided modular inclusions
Taking the wedge modular inclusion defined in Section 2.4.1 into account, we now prove that isotony can be deduced from covariance, the Bisognano-Wichmann property and the C-spectral condition. On specific models this has been checked in [BGL02,Lo08]. Proof. Let W 0 = (h, τ ) ∈ G E and H 0 = N(W 0 ). By covariance, the net N is isotone if and only if As the stabilizer G ↑ W0 stabilizes H 0 by covariance, isotony is equivalent to exp(x) ∈ S H0 for every x ∈ C + ∪ C − .

The Brunetti-Guido-Longo (BGL) construction
We have seen in the introduction to Section 2 that each standard subspace H specifies a homomor- Combining this with Φ leads to the so-called Brunetti-Guido-Longo (BGL) construction: This means that, with respect to Definition 2.3, U NU (W ) = U • γ W for W ∈ G (see [BGL02], [NÓ17,Prop. 5.6]). The BGL net associates to every wedge W ∈ G a standard subspace N U (W ). We shall denote with (W + , N U , U ) the restriction of the BGL net to the G ↑ -orbit W + ⊆ G E (G).
Theorem 4.12. The restriction of the BGL net N U associated to an (anti-)unitary C-positive representation U of G = G ↑ ⋊ {e, σ} to a G ↑ -orbit W + ⊆ G E satisfies all the axioms (HK1)-(HK3) and (HK5).
Proof. Let W + ⊆ G E (G) be a G ↑ -orbit. By construction, the restriction of the BGL net N U to W + satisfies (HK2) and by construction it satisfies (HK5). By Proposition 4.10, isotony (HK1) follows from the Spectral Condition (HK3), which is the C-positivity of U .
As a last remark in this section we stress that two (anti-)unitary extensions of a unitary representation (U, H) of G ↑ are unitarily equivalent, but the corresponding BGL nets depend on the choice of the (anti-)unitary extension. The following proposition makes this dependence explicit and provides a natural parameter space.

Twisted Locality
We have seen in Section 2.4.2 that it can happen that W ′ / ∈ W + = G ↑ .W . One can anyway attach to W ′ a real subspace by the BGL-net and by construction obtain the relation H(W ′ ) = H(W ) ′ . On the other hand one can define natural complementary wedges W ′ α indexed by central elements α.
In this section we will see that in the BGL construction, the complementary wedge subspaces satisfy the central Haag duality condition (HK6), hence the twisted locality relation (HK4). We start with a lemma on standard subspaces. Proof. The existence of the square root and the commutation relation with the modular conjugation and the modular operator follows by Lemma A.3. Then It is clear that H 1 does not depend on the choice of Z.
In order to conclude (HK6), hence the central locality condition on a BGL net N U , we will need an analogous statement relating complementary wedge subspaces. Proof. First we note that U (α) ∈ M := U (G ↑ ) ′ . We fix σ 0 ∈ Inv(G ↓ ) and observe that conjugation with U (σ 0 ) defines an antilinear isomorphism β of M. As β(U (α)) = U (α) −1 follows from α ∈ Z(G ↑ ) − , we find with Lemma A.3(c) in the appendix, a unitary square root Z α of U (α) satisfying For any other σ ∈ G ↓ we have σ = σ 0 g with g ∈ G ↑ , so that We are now ready to verify that the BGL net is compatible with the twistings appearing in (HK4), (HK6) and (HK7).
In this case JZ α J = Z * α holds for any antiunitary operator J. (b) Let (U, H) be an (anti-)unitary representation of G. For any other square root Z of U (α) satisfying the same requirements as Z α , the unitary operator Z −1 Z α is an involution commuting with U (G), so that it leaves all standard subspaces N(W ) of the BGL net invariant. (c) If α ∈ Z(G ↑ ) satisfies α σ = α for σ ∈ G ↓ , then α acts trivially on G(G) and, by covariance of N, leaves all standard subspaces N(W ) invariant. This happens in particular if α 2 = e. Then also α ∈ Z(G ↑ ) − , so that α-twisted complements are useful in the context of fermionic theories. Here U (α) is an involution and one choice of a square root of U (α) is given by Given a net satisfying (HK1)-(HK5), the commutation relation among twist operators and the wedge modular operators immediately hold.
In particular, the right hand side does not depend on the choice of Z α .
Proof. By (HK5), the unitary operator Z α ∈ U (G ↑ ) ′ commutes with the modular operator of N(W ), by Proposition 4.18. Therefore the two standard subspaces N(W ′ α ) and Z α N(W ) ′ have the same modular operator. By twisted locality N(W ′ α ) ⊆ Z α N(W ) ′ , so that Lemma 4.3 implies that they coincide. imply that it also has the correct commutation relation with exp(C ± ), hence also with G ↑ (W 0 ). We shall see in Section 4.4, when we actually obtain an extension to the full group G. be its simply connected covering. We write λ W for the one-parameter group lifting the boost group Λ W associated to a wedge W ∈ W = G.W 1 (see e.g. [Mo18]). For G ↑ , a wedge is defined by a pair W = (x, r x ), where x generates Λ W and r x = e πix is the spacetime reflection in the direction of the wedge. Since Z = Z(G) = {±1} is a 2-element group, a wedge W ∈ G has two lifts which belong to two different G ↑ -orbits in G(G). To see this, we note that Z = Z − and Z 2 = {e}. For the second equality we use the isomorphism Spin 1,3 (R) with SL 2 (C) and note that the centralizer of any Euler element x, which may be assumed to be x = , is connected and isomorphic to the multiplicative group C × , on which the involution σ x acts trivially. Therefore the central elements ∂(g) = g σ g −1 , g ∈ G ↑ (x,σx) , are all trivial, which leads to Z 2 = {e}. For α := −1, the twisted complement of W = (k W , σ W ) is W ′ −1 = (−k W , −σ W ). Any lift r : R → G ↑ of a rotation one-parameter group ρ : R → SO 2 (R) ֒→ SO 1,3 (R) in G ↑ satisfying Ad(ρ(π))k W = −k W now satisfies ρ(2π) = −1. This shows that,   2.10(e)). This group is obtained from Möb 2 by factorization of the subgroup nZ( Möb). Then Z := Z(G ↑ ) ∼ = Z n is a cyclic group of order n. Let α := ρ(2π) ∈ Z be a generator, where ρ : R → G ↑ is the lift of the rotation group.
Let (U, H) be an (anti-)unitary representation of G whose restriction to G ↑ is irreducible. Then, by Schur's Lemma, U (α n ) = U ( ρ(πn)) is an involution in T1, hence ±1. We now define n-twisted local nets of real subspaces as follows: • n is even. As β τ = β −1 for β ∈ Z, we have Z − = Z and Z 1 ∼ = Z n/2 is a subgroup of index 2.
As for Möb 2 , we have Z 2 = Z 1 . We therefore obtain for every Euler couple The locality relation then is given by where α = ρ(2πk) and ω ∈ T satisfies ω 2 1 = U ( ρ(2π)). Since U is irreducible and Z is a cyclic group of order n, U ( ρ(2π)) is an n-th root of the unity, hence ω 2n = 1 and Z α = ω k 1.
• n is odd. Then Z − = Z 1 implies that G ↑ acts transitively on the inverse images of G ↑ -orbits in G. Fixing the orbit G ↑ .(x, σ), we have by the BGL construction a net of real subspaces I → N(I), where, again, I is an interval of length smaller than 2π in the n-fold covering of S 1 . Here the locality relation is where α = ρ(2πk) and ω 2n = 1, I ′ α and Z α are as above.

New models
Theorem 3.10 provides the list of restricted root systems for real simple Lie algebras containing (symmetric) Euler elements, hence supporting (symmetric) Euler wedges. Any such Lie algebra g is the Lie algebra of a simply connected Lie group G ↑ . Then (2.10) defines an Euler involution on the group G ↑ , so that we obtain the extension to G = G ↑ ⋊ {id, σ}. Such a Lie group G ↑ has many unitary representations, possibly with positive energy if the Lie algebra g is hermitian. By unitary induction, one can construct a unitary representation of G ↑ from a unitary representation of a subgroup, for instance from a covering of PSL 2 (R) ⊂ G ↑ [Ma52]. It is always possible, to extend a unitary representation (U, H U ) of G ↑ to an (anti-)unitary representation of G by doubling the Hilbert space, if the representation does not extend on H U itself. Indeed, we can choose any conjugation C on H U and observe that the representation defined by U (g) = U (g) ⊕ CU (σgσ)C on H U ⊕ H U extends to G by U (σ) = 0 C C 0 . By the BGL construction there exists a (twisted-)local one-particle net satisfying (HK1-8).
As a consequence we have the theorem: In [GL95,Lo08] this theorem was proved for SL 2 (R)-representations of the principal and discrete series. Here the argument does not depend on the family of the representation.
Proof. Since SL 2 (R) is a type I group, every unitary representation has a unique direct integral decomposition into irreducible unitary representations. This reduces the problem to the irreducible case. We have to show that U • τ G ∼ = U * (the dual representation). Let Then h, k, u is a basis of sl 2 (R) and is a Casimir element, so that The antilinear extension τ of τ to sl 2 (R) satisfies τ (iu) = iu and the operator i∂U (u) is selfadjoint and diagonalizable. We have ∂U * (u) = −∂U (u) = ∂U (τ (u)), so that U * • τ G is an irreducible with the same u-weights and the same Casimir eigenvalue c. Below we argue that U is uniquely determined by any pair (µ, c), where µ is an eigenvalue of i∂U (u) occurring in the representation ( [Sa67], [Lo08]), and this implies that U • τ G ∼ = U * . To see that U is determined by the pair (µ, c), we first recall that H decomposes into onedimensional eigenspaces of i∂U (u) and, by irreducibility, it is generated by any eigenvector ξ µ of eigenvalue µ. Let U(g) denote the complex enveloping algebra of g. Then V µ := U(g)ξ µ is a dense subspace consisting of analytic vectors, so that the representation U is determined by the g-representation on this space. In U(g) the centralizer C u of u is generated by u and the Casimir element. Therefore ξ µ is a C u -eigenvector and the corresponding homomorphism χ : C u → C is determined by χ(u) = µ and χ(ω) = c. It is now easy to verify that these two values determine the U(g)-module structure on V µ , hence the unitary representation U .
Remark 4.25. Here the determination of the representation is obtained by considering in the enveloping algebra U(sl 2 (R)), the centralizer subalgebra C[ω, u] of u. Any cyclic weight vector ξ µ,c defines a character χ of this subalgebra by χ(iu) = µ and χ(ω) = c, and U(g)ξ µ,c is isomorphic to the quotient of U(g) by the left module generated by µ1 − iu and ω − c1.   The following theorem shows that an isotone, central twisted local G ↑ -covariant net of standard subspaces satisfying the BW property extends is actually G-covariant. The argument needs the density property described in Proposition 4.27 for SL 2 (R). The extension is done by (HK8).The proof generalizes the argument in [GL95].
Theorem 4.28. (Extension Theorem) Let G = G ↑ ⋊ {id, σ} be a graded Lie group, where σ is an Euler involution. Let (U, H) be a unitary C-positive representation of G ↑ , W + ⊆ G E (G) be a G ↑ -orbit, and (W + , N, U ) be a net of standard subspaces satisfying (HK1-4) and the BW property (HK5). If h 1 , . . . , h n , n ≥ 2, is a pairwise orthogonal family of Euler elements generating the Lie algebra g, and the conditions in Proposition 4.27 hold for the representations of the connected subgroups corresponding to the sl 2 -subalgebras generated by h 1 and h j for j = 2, . . . , n, then U extends to an (anti-)unitary representation of G such that G-covariance (HK7) and modular reflection (HK8) hold.
Proof. Let (W + , N, U ) be a net of standard subspaces satisfying (HK1-5). The Bisognano-Wichmann property (HK5) implies Central Haag Duality (HK6) by Proposition 4.19. Let H j := −i∂U (h j ) be the selfadjoint generators of the unitary one-parameter group corresponding to h j . By Corollary 3.14, every pair (h 1 , h j ) generates a subalgebra isomorphic to sl 2 (R) and the generators H 1 and H j integrate to a representation of SL 2 (R). Consider the Euler wedges W 1 , W j ∈ W + associated to h 1 and h j , respectively.
Since N coincides with the restriction to W + of the BGL net of the (anti-)unitary representation U of G, the representation U α satisfies (HK7) and (HK8) by Proposition 4.16.
Note that the density property as well as the existence of orthogonal wedges are sufficient but not necessary to have a G-covariant action: Consider the BGL net associated to the unique irreducible positive energy representation U of the G = Aff(R) on the real line. Then the standard subspaces N U (a, ∞) and N U (−∞, b) are associated to positive and negative half-lines and satisfy (HK1)-(HK5). There are no-orthogonal wedges in this case but the extension to an (anti-)unitary representation of G is given by U (σ W ) = J NU (W ) .
We further remarks that in this case σ W does not preserve the wedge family W + . For the Poincaré group, with the identification of wedge regions and Euler elements (see (2.28)), the axial wedges W j = {(t, x) ∈ R 1+d : |t| < x j }, j = 1, . . . , d, define a family of orthogonal wedge regions, namely wedge regions associated to orthogonal Euler elements. Considering wedges as subsets of Minkowski spaces one can define further regions by wedge intersection. Spacelike cones are particularly important: they are defined, up to translations by finite intersection of wedges obtained by Lorentz transforms of W 1 . Analogously one can define, by intersecting wedge subspaces, subspace associated to any spacelike cone. In principle this can also be trivial, but if they are standard, the cyclicity assumption of 4.27(c) is ensured, cf. [GL95]. Consider G = Möb ⋊ {id, τ }. Let (W + , U, N) be a net of standard subspaces satisfying (HK1)-(HK5). Let I ⊃ ⊆ R be an interval with q( I ⊃ ) = I ⊃ where the latter is the right semicircle with endpoints (−i, i) ⊂ S 1 . Then the dilation generators δ ∩ and δ ⊃ define orthogonal Euler elements generating Möb. Considering the wedges W ∩ = (x ∩ , σ ∩ ) and W ⊃ = (x ⊃ , σ ⊃ ) with W ∩ = ρ(π/2)W ⊃ , the intersection is again a wedge interval I = I ∩ ∩ I ⊃ . In particular, by isotony, N( I ∩ )∩N( I ⊃ ) ⊃ N( I) is standard and condition (c) in Proposition 4.27 holds.