E$_{6(6)}$ Exceptional Drinfel'd Algebras

The exceptional Drinfel'd algebra (EDA) is a Leibniz algebra introduced to provide an algebraic underpinning with which to explore generalised notions of U-duality in M-theory. In essence it provides an M-theoretic analogue of the way a Drinfel'd double encodes generalised T-dualities of strings. In this note we detail the construction of the EDA in the case where the regular U-duality group is $E_{6(6)}$. We show how the EDA can be realised geometrically as a generalised Leibniz parallelisation of the exceptional generalised tangent bundle for a six-dimensional group manifold $G$, endowed with a Nambu-Lie structure. When the EDA is of coboundary type, we show how a natural generalisation of the classical Yang-Baxter equation arises. The construction is illustrated with a selection of examples including some which embed Drinfel'd doubles and others that are not of this type.


Introduction
Dualities play an important role in our understanding of string theory. One of the bestunderstood dualities is T-duality, which relates string theory on backgrounds with U (1) d isometries, with the backgrounds related by O(d, d) transformations. These T-dualities are already visible in perturbative string theory, and are enlarged into U-dualities in the non-perturbative framework of M-theory [1,2]. Generalisations of Abelian T-dualities exist for backgrounds with non-Abelian isometries, leading to non-Abelian T-duality (NATD) [3], and for backgrounds without any isometries, called Poisson-Lie T-duality (PLTD) [4,5]. Instead of the isometry algebra, PLTD is controlled by an underlying Drinfel'd double.
Unlike Abelian T-duality, which is an equivalence between string theories on different backgrounds to all orders in the string coupling and string length, these generalised T-duality are currently best understood at the supergravity level and only to a limited extend beyond leading order in α [6] and their status as true dualities of the string genus expansion remains doubtful [7]. Nonetheless, NATD and PLTD have led to fruitful results. For example, NATD has been successfully used as solution-generating mechanisms of supergravity [8], leading to the discovery of new minimally supersymmetric AdS backgrounds starting with [9] (see [10] for a review and further references). Moreover, there is a close connection between PLTD and the (modified) classical Yang-Baxter equation which controls integrable deformations of σ-models [11,12].
The non-perturbative generalisation of Poisson-Lie T-duality to a U-duality version in Mtheory, or more conservatively as a solution-generating mechanism of 11-dimensional supergravity, has long been an open problem, which was recently addressed in [13,14] and further elaborated on in [15][16][17]. Building on the interpretation of PLTD and Drinfel'd doubles within Double Field Theory (DFT), [18][19][20], [13,14] used Exceptional Field Theory (ExFT)/Exceptional Generalised Geometry to propose a natural generalisation of the Drinfel'd double for dualities along four spacetime dimensions. This "Exceptional Drinfel'd Algebra" (EDA) was shown to lead to a new solution-generating mechanism of 11-dimensional supergravity that suggests a notion of Poisson-Lie U-duality, as well as a generalisation of the classical Yang-Baxter equation.
Other recent works [21][22][23] have considered closely related ideas, although the detailed relation between these approaches and the EDA is not completely apparent.
In this paper, we will further develop the ideas of [13,14] by constructing EDAs and Poisson-Lie U-duality amongst six directions. We choose six dimensions, because important new features arise when dualities are considered in six directions. This is because now the 6-form can completely wrap the six directions we are considering. As a result, the e 6(6) algebra contains a generator, corresponding to a hexavector, which will generate new kinds of dualities and deformations which have no counterpart in PLTD, as we will see.
The outline of the rest of this paper is as follows: in section 2 we describe the EDA from a purely algebraic perspective. In section 3 we show how the EDA can be realised within exceptional generalised geometry as a Leibniz parallelisation of a particular type of group manifold G, that we will call a (3, 6)-Nambu-Lie group. We then consider more closely the case of a coboundary EDA in section 4 whose structure is governed by a generalisation of the Yang Baxter equation. We provide a range of examples in section 5 of EDAs both coboundary and otherwise, some of which have Drinfel'd doubles as subalgebras, and other which do not have such an interpretation. The aim of these examples is not to provide here a full classification, which could form an interesting investigation in its own right, but rather to highlight the various features that can arise.
2 The E 6(6) EDA Before specialising to the case of E 6(6) we begin by presenting some generalities of the Exceptional Drinfel'd Algebra. The EDA, d n , is a Leibniz algebra which is a subalgebra of E n(n) 1 , admitting a "maximally isotropic" subalgebra, as we will define shortly. In table 1 we provide details of the representations of E n(n) inherited from the exceptional field theory (ExFT) approach to eleven-dimensional supergravity that are useful to the present construction.  The split real form of exceptional groups E n(n) with D = 11 − n, their maximal compact subgroups H n and representations R 1 . . . R 4 appearing in the tensor hierarchy of ExFT. In this work we will be mostly concerned with representations R 1 and R 2 which will be associated to the generalised tangent bundles E and N respectively.
We denote the generators of d n by {T A }, with the index A inherited from theR 1 representation of E n(n) ExFT and their product by with X AB C structure constants which are not necessarily antisymmetric in their lower indices.
The product obeys the Leibniz identity, namely which implies for the structure constants Note that if the Leibniz algebra is a Lie algebra, i.e. the X AB C are antisymmetric in their lower indices, then this reduces to the Jacobi identity.
We place two further (linear) requirements on the EDA. Firstly, we demand that there is a maximal Lie subalgebra g spanned by {T a } ⊂ {T A } obeying g ⊗ g|R 2 = 0 , (2.4) in which the representationR 2 is found in table 1. We call such a subalgebra g maximally isotropic. We will be interested here in the case that dim g = n as this is relevant to the Mtheory context. 2 Since G = exp g acts adjointly on d n , it follows that G should be endowed with a trivector and hexavector. We will further require that these objects give rise to a 3-and 6-bracket on g * , thereby imposing some further restrictions on the structure constants X AB C . 3 These additional requirements imply that the EDA can be given a geometrical realisation in terms of certain generalised frames whose action is mediated by the generalised Lie derivative (3.4), as we will show in section 3.
Let us now discuss these restrictions in detail.

Linear Constraints
We now study in detail the consequence of the requirements of the maximally isotropic subalgebra g and its adjoint action. Since these constraints arise from placing requirements directly to the form of X AB C we describe them as linear constraints; this is to be contrasted with quadratic constraints of the form X 2 = 0 that arise from the Leibniz identity.
Firstly, since g is a Lie algebra, we immediately have with f ab c antisymmetric in a, b. Secondly, the adjoint action 4 of g ∈ G = exp g on d n implies with (A g ) A B ∈ E 6(6) since g ⊂ d n ⊂ e 6(6) . Let us denote the adjoint action of g ∈ G on g by a g .
Then g · T A · g −1 takes the form: (2.7) 2 There is another inequivalent way to maximally solve the condition eq. (2.4) with dim g = n − 1 leading to a IIB scenario [24,25]. 3 It is worth emphasising that these are impositions beyond simply demanding that g be a maximal isotropic. 4 To be more precise we inherit an action via the rack product: Thus, G admits a totally antisymmetric trivector λ abc and totally antisymmetric hexavector λ a 1 ...a 6 which control its adjoint action on the generators T ab and T a 1 ...a 5 .
(λ e ) abc = (λ e ) a 1 ...a 6 = 0 , (2.8) and they inherit a group composition rule Finally, we come to the second condition on the EDA, i.e. the existence of a 3-and 6-bracket on g * . This is equivalent to imposing the following differential conditions on λ abc and λ a 1 ...a 6 : where r = r a T a are the right-invariant 1-forms on G obeying dr a = 1 2 f bc a r b ∧ r c and we have dropped the subscript g on λ (3) and λ (6) . The f b a 1 ...a 3 and f b a 1 ...a 6 are structure constants for a 3-and 6-bracket and are totally antisymmetric in their upper indices. In fact, as we will see in section 2.2, the Leibniz identity implies further properties of the trivector and hexavector, in particular that they define a certain Nambu 3-and 6-bracket which are compatible with the Lie bracket on G. Therefore, it seems apt to call G a (3,6)-Nambu-Lie Group.
With the above conditions, the EDA takes the following form (2.11)

Leibniz identity constraints
We will now study the compatibility conditions between the Lie algebra g, the 3-bracket and 6-bracket, as well as their appropriate "closure" conditions that are required for the EDA to satisfy the Leibniz identity of eq. (2.2). This yields a number of immediate constraints. In particular, we obtain the following fundamental identities, i.e. generalisations of Jacobi for higher brackets, as well as compatibility conditions between the dual structure constants f b a 1 a 2 a 3 , f b a 1 ...a 6 and the Lie algebra structure constants. These compatibility conditions take the form of cocycle as well as the additional constraint If we only consider EDAs d n with n ≤ 6, as we are doing here, the conditions given by the above eqs. (2.12)-(2.17) are equivalent to imposing the Leibniz identity. This is because in n ≤ 6, the fundamental identity for the six-bracket implies that f b a 1 ...a 6 = 0. However, since the structure we are studying here will also exist for n > 6, we will keep the remaining discussion as dimension-independent as possible, whilst keeping in mind that for n > 6, the Leibniz identity will lead to further or modified compatibility conditions between f ab c , f b a 1 a 2 a 3 and f b a 1 ...a 6 .
These additional constraints will need to be studied using EDAs based on E 7(7) and higher.
Before interpreting these constraints, we remark that the Leibniz identity ensures, much as the structure constants of a Lie algebra g are invariant under G = exp g acting adjointly, that the EDA structure constants enjoy an invariance Substitution of eq. (2.7) here results in a variety of identities that we shall revisit later on.

Fundamental identities
Let us now introduce the 3-bracket { } 3 and 6-bracket { } 6 on g * with structure constants f b a 1 a 2 a 3 and f b a 1 ...a 6 , respectively, i.e.
{x, y, The conditions (2.12) and (2.13) imply that the 3-and 6-brackets satisfy for all x 1 , . . . , x 5 , y 1 , . . . , y 6 ∈ g * , and where we used the Lie bracket on g to define the ad- which is given, assuming a basis {T a } for g * , by We see that the 6-bracket must satisfy the fundamental identity for Nambu 6-brackets, while the 3-bracket's fundamental identity is modified by the 6-bracket and the co-product defined by the structure constants of g.

Compatibility conditions
The first set of compatibility conditions, eqs. (2.15) and (2.16), between the 3-and 6-brackets and the Lie algebra g imply that f b a 1 a 2 a 3 defines a g-cocycle and that f b a 1 ...a 6 is an f 3 -twisted g-cocycle, as follows. f b a 1 a 2 a 3 and f b a 1 ...a 6 define Λ 3 gand Λ 6 g-valued 1-cochains (2.24) Using the coboundary operator d : g * ⊗ Λ p g −→ Λ 2 g * ⊗ Λ p g, for p = 3 and p = 6 here, The coboundary operator is nilpotent with d : for all x ∈ g and ρ p ∈ Λ p g. Therefore, the cocycle conditions (2.27) can be solved by the (twisted) coboundaries In components, these are equivalent to The coboundary case is related to a generalisation of Yang-Baxter deformations. The trivector ρ a 1 a 2 a 3 and the hexavector ρ a 1 ...a 6 correspond to the M-theoretic analogue of the classical rmatrix. The equations corresponding to the classical Yang-Baxter equations for the r-matrices are implied by substituting the solutions (2.30) to the fundamental identities (2.13) and (2.14).
We will discuss this further in section 4.
Finally, the additional constraint (2.17) implies that the ad-invariant co-product ∆ on g * (2.21) defines a commuting subspace of the 3-bracket: 3 E 6(6) EDA from generalised frame fields We now provide a geometric realisation of the E 6(6) EDA by constructing a Leibniz parallelisation [26][27][28][29][30][31] of the exceptional generalised tangent bundle [32][33][34][35][36][37][38] in which we identify the manifold M = G = exp g. We will also be interested in a second bundle The action of sections of these bundles, is mediated by the generalised Lie derivative [34][35][36] defined as We define [35] a symmetric bilinear map ·, · : E × E → N as such that the generalized Lie derivative satisfies The parallelisation consists of a set of sections E A ∈ Γ(E) that: • form a globally defined basis for Γ(E) • give rise to an E 6(6) element 5 , E A M , whose matrix entries are the components of E A • realise the algebra of the EDA through the generalised Lie derivative where the constants X AB C are the same as those defined through the relations in eq.
The parallelisation can be directly constructed in terms of the right-invariant Maurer-Cartan one-forms on G, r a , their dual vector fields e a , and the trivector, λ a 1 a 2 a 3 , and hexavector, λ a 1 ...a 6 .
This can thus be thought of as a special example of the more general prescription of [39], where we only make use of the aforementioned geometric data on the (3, 6)-Nambu-Lie Group G.
Following the decomposition of EDA generators we write (3.9) It is straightforward, but indeed quite lengthy, to verify that these furnish the EDA algebra.
A first check is to see that after using the identities (2.10) to evaluate derivatives we can go to the identity of M where λ a 1 a 2 a 3 and λ a 1 ...a 6 vanish. One then has to use the adjoint invariance conditions that follow from eq (2.18) to conclude that this holds away from the identity.
If we specialise now to the case of f b a 1 ...a 6 = 0, which we recall is enforced for n ≤ 6 by the fundamental identities, we find quickly an immediate consequence of eq. (2.18) is that dλ a 1 ...a 6 = 0, and since λ a 1 ...a 6 vanishes at the identity, it must be identically zero. The remaining adjoint invariance conditions can be combined to imply that These conditions are sufficient to ensure that frame algebra is obeyed.
We also define the generalized frame field E A , which is a section of N , through where η AB C is an invariant tensor of the E n(n) . For E 6(6) this tensor is related to the symmetric invariant (see the appendix for details) such that the explicit form of E A has components E a ,a 1 ...a 6 = (λ a 1 ...a 6 r a − 30 λ a [a 1 a 2 λ a 3 a 4 a 5 r a 6 ] ) − 15 λ a [a 1 a 2 r a 3 ...a 6 ] + jr a r a 1 ...a 6 . (3.12) Here we denote r a 1 ...am = r a 1 ∧ · · · ∧ r am and make use of the j-wedge contraction of [32] to deal with mixed symmetry fields. 6 One can consider now the action of the frame field E A on these E A and by virtue of eq. (3.11), again find that they furnish the EDA algebra, albeit in a different representation as described in the appendix.

Generalised Scherk-Schwarz reductions and the Embedding Tensor
The generalised frame field introduced above can be used as a compactification Ansatz within ExFT known as a generalised Scherk-Schwarz reduction. In this procedure all internal coordinate dependence is factorised into dressings given by the generalised frame. The algebra in 6 For a p + 1-form α and a (n − p)-form β, we define 13) eq. (3.8) ensures the dimensional reduction results in a lower dimensional gauged supergravity.
The structure constants of the EDA determine the gauge group of this lower dimensional theory, and in such a context are known as the embedding tensor. To facilitate contact with the literature [40] we can express this in terms of the 27 and 351 representations of E 6(6) as 14) The components of the antisymmetric Z AB are determined to be and those of ϑ A (sometimes called the trombone gauging) to be This suggests a natural generalisation of Yang-Baxter deformations to EDAs [13,14]. We begin with an EDA d 6 with only the structure constants, f ab c , corresponding to a maximally isotropic Lie subalgebra g, non-vanishing and f a bcd = f a b 1 ...b 6 = 0, i.e.
We denote the structure constants collectively asX AB C .
We now perform an E 6(6) transformation of the above EDA by a trivector, ρ abc , and hexavector, ρ a 1 ...a 6 , which will play the analogue of the classical r-matrix. The corresponding E 6(6) group element is given by in which the generators R a 1 a 2 a 3 and R a 1 ...a 6 are specified in the appendix. Explicitly we have Equivalently, we twist the generators by the group element (4.2) resulting in For the twisted generators, we obtain We now require that the new algebra defines an EDA d 6 . This imposes conditions on ρ abc and ρ a 1 ...a 6 and we will interpret these as analogues of the classical Yang-Baxter equation. From the products T a • T b 1 b 2 and T a • T b 1 ...b 5 , the dual structure constants are identified as which take the form of (twisted) coboundaries (2.30) and the (= 0) holds for d 6 . The remaining products impose a number of conditions, of which the following is particularly intriguing This is a natural generalisation of the classical Yang-Baxter (YB) equation which we will elaborate more on later. In general, we get a further set of conditions which are required to ensure that the new algebra defines an EDA d 6 . Mostly these additional conditions appear rather cumbersome but we note the requirement that ρ a 1 a 2 b f a 1 a 2 c = 0 . In [21], a different approach was taken using a generalisation of the open/closed string map to propose a generalisation of the classical YB equation for a trivector deformation of 11-dimensional supergravity. The approach of [21] is not limited to group manifolds, unlike the present case, but also only considers trivector deformations. However, when specialising [21] to group manifolds and considering our deformations with ρ a 1 ...a 6 = 0, the resulting equation of [21] is different and, in particular, weaker than the YB equation we find here (4.8) with ρ a 1 ...a 6 = 0, or indeed the SL(5) case discussed in [13,14]. Indeed, as shown in [22] based on explicit examples, the proposed YB-like equation of [21] is not sufficient to guarantee a solution of the equations of motion of 11-dimensional supergravity, while our deformations subject to the above conditions preserve the equations of motion of 11-dimensional supergravity by construction.

The generalised YB equation
To understand better the generalised YB equation obtained above, let us adopt a tensor product Introducing a (anti-)symmetrizer in the tensor product σ [123], [45] , allows this equation to be concisely given as [45] [ρ 123 + ρ 234 , ρ 145 ] = 0 . Suppose that we have a preferred q ∈ g such that with r 12 neutral (i.e. [r 12 , q 1 ] = 0) then we find eq. (4.17) becomes    where, for example, Pictographically this is indicated in figure 1.

Examples
In this section we wish to present a range of examples of the EDA, both of coboundary type and otherwise. We will give some broad general classes that correspond to embedding the algebraic structure underlying existing T-dualities of the type II theory. In addition, in the absence of a complete classification, here we provide a selection of specific examples.

Abelian
When the subalgebra {T a } is Abelian, arbitrary ρ a 1 a 2 a 3 and ρ a 1 ...a 6 are solutions of Yang-Baxterlike equations. However f a b 1 b 2 b 3 = 0 and f a b 1 ...b 6 = 0 and the EDA is Abelian.

Semi-Abelian EDAs and Three Algebras
The algebraic structure corresponding to non-Abelian T-duality is a semi-abelian Drinfel'd double i.e. a double constructed from some n−1 dimensional Lie-algebra (representing the non-Abelian isometry group of the target space) together with a U (1) n−1 (or perhaps R n−1 + ) factor.
An analogue here would be to take f ab c = 0 and f a b 1 ...b 3 = 0, this however is not especially interesting. More intriguing is to consider the analogue of the picture after non-Abelian Tdualisation has been performed in which the U (1) n−1 would be viewed as the physical space.
This motivates the case of semi-Abelian EDAs with f ab c = 0 but f a b 1 ...b 3 = 0.
In this case the Leibniz identities reduce to the fundamental identities Each solution for this identity gives an EDA. To identify these one can use existing classification efforts and considerations of three algebras that followed in light of their usage [47] to describe theories of interacting multiple M2 branes.
The first case to consider are the Euclidean three algebras, such that totally antisymmetric. Here the fundamental identity is very restrictive and results in a unique possibility: the four-dimensional Euclidean three algebra [48][49][50], whose structure constants are just the antisymmetric symbol, complemented with two U (1) directions. Relaxing the requirement of a positive definite invariant inner product allows a wider variety [51][52][53][54][55][56]. Dispensing the requirement of an invariant inner product (which thus far appears unimportant for the EDA) allows non-metric three algebras [57][58][59]. 8

r-matrix EDAs
We now consider coboundary EDAs given in terms of an r-matrix as in eq.(4.19) obeying the YB equation (4.21). Splitting the generators of g into Tā withā = 1, . . . , 5 and T 6 (identified with the generator q appearing in (4.19)) we have the non-vanishing components ρāb 6 = rāb.
Furthermore the condition (4.9) requires that rābfāb 6 = rābfābc = rābfā 6c = rābfā 6 6 = 0 , (5.2) in which the last two equalities match the statement that r is neutral under T 6 . In such a setup, the dual structure constants are specified as Assuming further thatḡ = span(Tā) is a sub-algebra of g then rāb defines an r-matrix onḡ obeying the YB equation. Consequentlyfb 1b2ā = −2fāc [b 1 r |c|b 2 ] are the structure constants of a dual Lie algebraḡ R andd =ḡ ⊕ḡ R is a Drinfel'd double. Thus we have a family of embeddings of the Drinfel'd double into the EDA specified by fā 6 6 and fā 6b . When g =ḡ ⊕ u(1) is a direct sum (such that fā 6 6 = fā 6b = 0), then this is precisely an example of the non-metric three algebra of [57][58][59]. We emphasise though that not every (coboundary) double can be embedded in this way; one must still ensure that equation (5.2) holds.

Explicit examples
We now present a selection of explicit examples that illustrate coboundary and non-coboundary EDAs.

Trivial non-examples based on SO(p, q)
To illustrate that the EDA requirements are indeed quite restrictive we can first consider the case of g = so(p, q) with p + q = 4. A direct consideration of the Leibniz identities reveals that there is no non-zero solution for f a b 1 ...b 3 (in fact the cocycle conditions alone determine this).
Equally the Leibniz identities admit only trivial solutions in the case of iso(p, q) = so(p, q) R p+q + with p + q = 3.

An example both coboundary and non-coboundary solutions
We consider an indecomposable nilpotent Lie algebra We find a family of solutions This means that only when d i (i = 1, . . . , 5) have the form and satisfy d 1 = 0 or d 3 = 0, the cocycle becomes the coboundary. 9 Here we introduced a parameter c0 for convenience, which is 1 in [62].

An example with ρ 6
In the previous example ρ 6 is absent. By considering the Lie algebra of the form g = g 4 ⊕ u(1) ⊕ u(1), where g 4 denotes a real 4D Lie algebra that is classified in [63], one can construct a number of examples 10 (based on unimodular Lie algebras) that admit ρ 6 . To illustrate this let us consider the case that g 4 = A 4,1 specified by structure constants We find the generalised Yang-Baxter and compatibility equations admit the following family of solutions: (5.10) The corresponding dual structure constants are (5.11)
Supposing αβ = 0, we find the general solution for ρ abc is given by where c 1 = 0 when α = β . The corresponding dual structure constants are  This solution contains an r-matrix EDA as a particular case c 1 = c 3 = 0 . 10 In the notation of [63] examples with ρ 6 = 0 are found when g4 is one of the following: A3,1 +u (1)

Conclusion and Outlook
In this work we have consolidated the exploration of exceptional Drinfel'd algebras introduced in [13,14] extending the construction to the context of the E 6(6) exceptional group. The algebraic construction here requires the introduction of a new feature: we have to consider not only a Lie algebra g together with a three-algebra specified by f 3 ≡ f a b 1 ...b 3 as in [13,14], but we have to also include a six-algebra f 6 ≡ f a b 1 ...b 6 . The Leibniz identities that the EDA must obey enforce a set of fundamental (Jacobi-like) identities for the three-and six-algebra as well as some compatibility conditions. These compatibility conditions require that f 3 be a g-cocycle and f 6 be an f 3 -twisted g-cocycle. In terms of the g coboundary operator d this can be stated as We can solve this requirement with a coboundary Ansatz, f 3 = dρ 3 and f 6 = dρ 6 + 1 2 ρ 3 ∧ dρ 3 , reminiscent of the way a Drinfel'd double can be constructed through an r-matrix. Indeed, we find a generalised version for the Yang Baxter equation for ρ 3 , concisely expressed as [45] [ρ 123 + ρ 234 , ρ 145 ] = 1 2 (ρ 12345;5 + ρ 12345;4 ) .
We proposed a 'quantum' relation from which this classical equation can be obtained. This feature, and the resultant interplay between one-, three-, and six-algebras, opens up many interesting avenues for further exploration.
The construction of the EDA is closely motivated by considerations within exceptional generalised geometry. We have shown how the EDA can be realised as a generalised Leibniz parallelisation of the exceptional generalised tangent bundle of a group manifold G. The data required to construct this mean that G is equipped with a 3-bracket and a 6-bracket which invites the consideration of Nambu-Lie groups.
Now we come to solving the various constraint equations that govern the structure of the EDA. The first thing to note is that due to the dimension, the only solutions to the fundamental identities have vanishing f 6 (and consequently a trivial 6-bracket on G). We believe however that in higher dimension this condition is less stringent and that there will solutions for which the structure described above is exhibited in full.
We There are several exciting open directions here that we share in the hope that others may wish to develop them further: • Extensions of the EDA to E 7 (7) and higher are likely to shed further light on the structures involved. As the space gets larger there is more scope to find interesting solutions.
• It would be interesting to develop a more general classification of EDA solutions.
• One feature of the EDA is that they may admit multiple decompositions into physical spaces, and a resultant notion of duality. Further development should go into this very interesting aspect.
• Here we make some robust requirements that result in structures compatible with maximally supersymmetric gauged supergravities. It would likely be interesting to see how the requirements of the EDA can be consistently relaxed to lower supersymmetric settings, for example using [65].
• On a mathematical note perhaps the most intriguing area of all is to develop the 'quantum' equivalent of the classical EDA proposed here. A E 6(6) × R + Algebra and conventions

Acknowledgements
The matrix representations of the E n(n) generators (t α ) A B in the R 1 -representation are This plays a similar role to that of the O(d, d) invariant inner product of generalised geometry and indeed can be used to construct the Y-tensor [36] given by Y AB CD = η AB E η CD E = 10 d ABE d CDE that is ubiquitous in exceptional field theory/generalised geometry.