N ov 2 01 9 U-duality extension of Drinfel ’ d double

A family of algebras En that extends the Lie algebra of the Drinfel’d double is proposed. This allows us to systematically construct the generalized frame fieldsEA I which realize the proposed algebra by means of the generalized Lie derivative, i.e., £̂EAEB I = −FAB C EC I . It turns out that the generalized frame fields include a twist by a Nambu–Poisson tensor. A possible application to the non-Abelian extension of U -duality is also discussed.


Introduction
The familiar T -duality is a symmetry of string theory when the target space has commuting (or Abelian) Killing vectors. An extension of the T -duality, where the Killing vectors do not commute with each other, has been proposed in [1][2][3][4][5][6], and it is known as non-Abelian T -duality (see [7] for a list of references). Subsequently, a further extension, called the Poisson-Lie (PL) T -duality has been found in [8,9], and it can be applied to a more general class of target spaces.
As it has been discovered there, there is a group structure of the Drinfel'd double behind the PL T -duality, and the symmetry of the PL T -duality can be understood as a freedom in the choice of the physical subgroup G out of the doubled Lie group.
In the case of Abelian T -duality, the double extension of the target space has been proposed in various contexts (see for example [10][11][12][13][14]). The geometry of the doubled space has been studied in [15][16][17], and more recently, the idea has been developed in the context of double field theory (DFT) [18,19]. In the original formulation of DFT, the symmetry of Abelian T -duality is manifest, but the non-Abelian T -duality or the PL T -duality has not been clearly discussed.
A new formulation of DFT on group manifolds (called DFT WZW ) has been developed in [20][21][22], and in the recent works [23,24], the PL T -duality has been studied in the framework of DFT WZW . In more recent papers [7,25], the non-Abelian T -duality and the PL T -duality have been discussed by using another approach, called the gauged DFT [26][27][28][29][30][31] (see also [32,33] for recent discussion on the Drinfel'd double and related aspects in DFT). Thus, DFT is now not restricted to Abelian T -duality but can be applied also to the non-Abelian extensions.
When the target space has D Abelian Killing vectors, the T -duality group of type II superstring theory is O(D, D). As it is well-known, this T -duality group is only a subgroup of a larger duality group, called the U -duality group. The U -duality group is E n (n ≡ D + 1), which is summarized in Table 1.1. In order to manifest the U -duality symmetry, the doubled space is not enough. As it has been discussed in [34][35][36][37][38][39], we need to extend the n-dimensional space (with Abelian Killing vectors) into an extended space with dimension D n , which is the dimension of the vector representation R 1 of the U -duality group (see Table 1.1). For higher n, it is much larger than the doubled space with dimension 2D, and the extended space is called the exceptional space for the obvious reason. The U -duality-manifest formulation of  Table 1.1: The U -duality group E n for each n and dimensions of two representations, known as the R 1 -representation and the R 2 -representation.
supergravities have been studied in [36,[39][40][41][42][43][44][45][46][47][48][49], and more recently, a formulation using a similar language to DFT has been developed in [50][51][52][53][54][55][56][57], which is called the exceptional field theory (EFT). Similar to the case of DFT, for the consistency of the theory, we need to choose a physical subspace from the extended space, and all of the supergravity fields are defined on the physical subspace. In the case of DFT, the maximal dimension allowed for the consistency is always D-dimensional, but in the case of EFT, there are two maximal choices, n-dimensions and D(= n − 1)-dimensions [58]. If we adopt the former choice, the target space of M-theory is reproduced while the latter reproduces that of type IIB theory [58]. In this sense, EFT unifies the geometry of M-theory and type IIB string theory, and the structure of the exceptional space is much richer than that of the double space.
Unlike the case of T -duality, there is no concrete proposal for the extension of U -duality when the Killing vectors are non-Abelian. Originally, the non-Abelian T -duality has been discovered by introducing certain gauge fields (associated with the Killing vectors) into the string sigma model. This allows us to reformulate the string theory as a gauged sigma model, which reduces to the standard string sigma model if we first eliminate certain auxiliary fields.
On the other hand, if we eliminate the gauge fields first, the string sigma model on the dual geometry is recovered [3][4][5][6], and in this sense, the gauged sigma model connects the original geometry and the dual geometry. If we try to apply the same procedure to the membrane sigma model, we face a difficulty (see [7]). In order to formulate the non-Abelian extension of U -duality, the approach of the PL T -duality will be more useful. For this purpose, we need to extend the exceptional space to some extended group manifold, similar to the Drinfel'd double. Such an extension has been studied in [59], but the relation to the Drinfel'd double is not so clear and no proposal has been made for the extension of the PL T -duality.
In this paper, by using the idea of EFT, we propose a family of Leibniz algebra E n which contains the Lie algebra of the Drinfel'd double as a subalgebra in a particular case. For simplicity, we restrict our analysis to the case n ≤ 4 . In that case, the generators of the algebra T A (A = 1, . . . , D n ) can be parameterized as {T A } = {T a , T a 1 a 2 }, where a = 1, . . . , n and T a 1 a 2 = −T a 2 a 1 . The algebra can be expressed as , and the following bilinear forms are defined: which naturally extend the standard bilinear form ·, · of the Drinfel'd double.
By using the proposed algebra, we can systematically construct the generalized frame fields where F AB C is the structure constant of the proposed Leibniz algebra Here,£ V is the generalized Lie derivative in EFT, which generates the gauge transformations in EFT. The systematic construction of E A I is indispensable to perform the PL T -duality or its extension PL T -plurality [60], and we expect that the U -duality extension presented in this paper will be useful in studying the non-Abelian extension of U -duality.
The structure of this paper is as follows. In section 2, we briefly summarize the idea of the PL T -duality. In particular, we explain how the relation (1.3) is important in the PL T -duality.
In section 3, we find the algebra E n and study its detailed properties. The construction of the generalized frame fields is explained in section 4. In section 5, we show several examples of the E n algebra. Section 6 is devoted to the summary and discussion.

Poisson-Lie T -duality
In this section, we review the PL T -duality by using the language of DFT.
Basics definitions in DFT: Let us set up basic definitions of DFT. We consider a doubled space which has the generalized coordinates (x M ) = (x m ,x m ) (m = 1, . . . , D). The metric and the B-field are packaged into the generalized metric, 1) and the dilaton Φ is redefined into the T -duality-invariant combination, and use these to raise or lower the indices M, N . The fields H M N (x) and d(x) are formally defined on the doubled space, but for the consistency, we impose the section condition, for arbitrary fields A(x) and B(x) . According to the section condition, all of the fields can depend only on a set of D coordinates, and in this paper, we choose x m as such D coordinates.
Any other choices can be mapped to this choice by performing a T -duality transformation.
A generalization of the Lie derivative, called the generalized Lie derivative is defined aŝ  Abelian T -duality: Now, let us consider the T -duality by using the above notation. In order to perform the standard Abelian T -duality, the generalized metric and the DFT dilaton are required to be constant, in a certain adapted coordinate system. In this constant background, equations of motion of DFT are trivially satisfied, and a constant O(D, D) transformation maps the solution to another constant solution.
Poisson-Lie T -dualizable backgrounds: When we consider the PL T -duality, the target space is allowed to be non-constant. The generalized metric H M N can be twisted by a nonconstant matrix E M A (x) and the DFT dilaton also can have a non-constant factor, where |ℓ(x)| ≡ |det(ℓ a m )| and the matrices, E M A (x) and ℓ a m (x) are defined as follows. First, we introduce the Lie algebra of the Drinfel'd double, which is equipped with the ad-invariant bilinear form, The indices A, B are raised or lowered by using the metric η AB . The structure constants f ab c andf ab c can be chosen arbitrarily as long as Jacobi identities are satisfied. Secondly, we decompose the algebra into two subalgebras g ⊕g, where g is the "physical algebra" spanned by T a and the dual algebrag is spanned byT a . Each of these is maximally isotropic for the bilinear form ·, · . We then define a group element g by using the generators of the physical subalgebra, for example, g = e x a Ta , and define the left-and right-invariant forms as (2.12) The left-and right-invariant vectors are denoted by v m a and e m a (ℓ a m v m b = δ a b and r a m e m b = δ a b ). We also parameterize the adjoint action of g −1 on the generators of the Drinfel'd double as Finally, by using the above quantities, we define the twist matrix as (2.14) Once the structure constants f ab c andf ab c and the parameterization of g are given, the matrices E M A (x) and ℓ a m (x) are uniquely obtained. Then, by using these matrices, the PL T -dualizable background is expressed as (2.9). For later convenience, it is useful to note that at the identity g = 1 (which corresponds to x a = 0 when g = e x a Ta ), we have by their definitions. We also note that the Abelian T -dualizable background, i.e., the constant background (2.7), is reproduced as a particular case, f ab c = 0 andf ab c = 0 with g = e x a Ta .
Poisson-Lie T -duality: We denote the Lie algebra of the Drinfel'd double as where For simplicity, we here suppose that the structure constant of the dual algebra is unimodular f ba b = 0 . 1 Then, the equations of motion of DFT for a general PL T -dualizable background (2.9) reduce to the following algebraic equations: This is the PL T -duality, which extends the Abelian T -duality (2.8). 3 In DFT, the tensor F AB C is called the generalized flux, and it is generally defined aŝ Then, by using the new twist matrix E ′ A M and the relation (2.9), the metric, B-field, and the dilaton in the dual geometry are obtained as Then, by decomposing the new generators as (T ′ A ) = (T ′ a ,T ′a ), we obtain a new physical subalgebra g ′ spanned by T ′ a . By parameterizing a group element as before, such as g ′ ≡ e x ′a T ′ a , we can again obtain the matrices Π ′ab , ℓ ′a m , and r ′a m . Then, the new generalized frame fields,  3 To be more precise, the PL T -duality is a particular transformation (CA B ) = 0 1 D The algebra of the Drinfel'd double provide a systematic way to construct the generalized frame fields satisfying (2.21), and the dual geometry can be explicitly constructed.
In the next section, we introduce an extension of the Drinfel'd double and explain a systematic way to construct a generalized frame fields satisfyinĝ by means of the generalized Lie derivative in EFT.

Leibniz algebra based on U -duality
Here, we propose a Leibniz algebra E n by using the generalized Lie derivative in the E n EFT.
For this purpose, let us begin with a quick introduction to EFT.
Basic definitions in EFT: As we have explained in the introduction, in EFT, we introduce an exceptional space with dimension D n . When we adopt the M-theory picture, we decompose the generalized coordinates x I (I = 1, . . . , D n ) as [35] ( where the multiple indices are totally antisymmetric and the numerical factors are introduced for convenience. The ellipses are not necessary as far as we consider the cases n ≤ 6. The supergravity fields such as the metric and gauge potentials are contained in the generalized metric M IJ , which extends the one H M N in DFT. The fields are formally defined on the exceptional space, but the extension of the section condition (2.4) again restricts the coordinate dependence. In order to reproduce M-theory, we choose x i as the physical coordinates and any more coordinate dependence is not allowed by the section condition. Thus, in the following discussion, we eliminate the coordinate dependence on the dual coordinates, Similar to DFT, the generalized Lie derivative in EFT is defined as [51] where Y IJ KL is an invariant tensor satisfying£ V Y IJ KL = 0 . For our purpose, it is enough to know the expression under the situation where all fields depend only on the physical coordinates x i .
In that case, the generalized Lie derivative is expressed as (see [61] for our convention) where the two arbitrary generalized vectors V I and W I are parameterized as and we have defined v p ≡ 1 p! v i 1 ···ip dx i 1 ∧ · · · ∧ dx ip and similar for w p . We note that the expression (3.4) coincides with the Dorfman derivative in generalized geometry [47,49].
In order to simplify our discussion, we restrict our attention to n ≤ 4 . Then, terms with five (or more) antisymmetrized indices identically vanish (e.g. v i 1 ···i 5 = 0) and the above generalized vectors reduce to Generalized frame fields in EFT: In order to consider the relation (2.20) in EFT, let us introduce certain generalized frame fields E A I in EFT. By considering the analogy with the DFT case (2.24), we consider the following parameterization: , and e m a (or r a m ) is a certain right-invariant vector (or 1-form) satisfying Then, using (3.6), we can compute the generalized Lie derivative as follows: and D a ≡ e i a ∂ i . In general, the generalized flux X AB C is not constant. 4 Unlike the DFT case, the first two indices are not antisymmetric X AB C = X [AB] C and even if we find a certain situation where X AB C is constant, the algebra is not a Lie algebra.
Accordingly, in the following, we investigate a Leibniz algebra satisfying is satisfied similar to the case of the generalized Lie derivative, Construction of the algebra E n : Here we take a heuristic approach to find the Leibniz algebra E n . First, we suppose that the generalized flux X AB C is constant, and assume that there exists an algebra (3.18) with F AB C = X AB C . Secondly, we assume that Π a 1 a 2 a 3 = 0 at a certain point x a = 0 , which corresponds to (2.15). We further assume that the so-called R-flux X a 1 a 2 b 1 b 2 c vanishes. At least when e i a = δ i a , this requirement is precisely a condition for Π a 1 a 2 a 3 to be a Nambu-Poisson tensor [64], and the condition X a 1 a 2 b 1 b 2 c = 0 will be understood as a natural generalization of the definition of the Nambu-Poisson tensor. In the case of the Drinfel'd double, the bi-vector Π ab has been a Poisson tensor, and in our setup, the Poisson tensor is naturally extended to the Nambu-Poisson tensor.
Under these assumptions, the generalized flux X AB C at the point x a = 0 reduces to This suggests us to define a new Leibniz algebra E n as for the generators T a and T a 1 a 2 requires the following relations:

35)
A bilinear form: We also introduce the bilinear form, which extends the bilinear form ·, · of the Drinfel'd double. A natural extension of the bilinear form has been known in EFT, 5 where η IJ; K connects a product of two R 1 -representation and another representation, called the R 2 -representation (see for example [51]) whose dimension d n is given in Table 1.1. Namely, the additional index K appended to the bilinear form transforms in the R 2 -representation. This index can be decomposed as (see [61] for the explicit form of η IJ; K ) and in our case n ≤ 4 , it is enough to consider the first two components, The bilinear form takes the form for arbitrary two vectors V I and W I parameterized as (3.6). Under an arbitrary E n U -duality transformation Λ, the tensor η IJ; K behaves as where Λ I J and Λ I J denote the same E n transformation in the R 1 -and R 2 -representation, respectively. Now, we introduce a matrix, which satisfies and redefine the bilinear form as Then, the bilinear form for the generalized frame fields (3.7) becomes Identifying the generalized frame fields E A with the E n generator T A , we define the following bilinear form for the generators: We note that the subalgebra spanned by {T a } is maximally isotropic for the bilinear form.
In fact, the isotropicity shows that the subalgebra is a Lie algebra This can be understood from the explicit form of the generalized Lie derivative (3.6), namely, When V I and W I satisfy V, W A = 0 , we have 48) and the generalized Lie derivative satisfies£ V W I = −£ W V I . Accordingly, for a set of the generalized frame fields {E a } forming an isotropic subalgebra, we havê 49) and the subalgebra is a Lie algebra. This property plays an important role when we explicitly construct the generalized frame fields.
E n generators: For the sake of clarity, let us explain our convention for the E n generators.
We decompose the E n generators {tα} (α = 1, . . . , dim E n ) for n ≤ 4 as [40] ( Their matrix representations (tα) A B in the R 1 -representation are given as follows: 6 (3.51) The matrix representations (tα) A B in the R 2 -representation are (3.52) Now, let us rewrite the E n algebra. If we express the algebra as 53) 6 The second term in the GL(n) generator (K a b )A B , which is proportional to the identity matrix δ B A , is necessary for the commutator [R c 1 c 2 c 3 , R d 1 d 2 d 3 ] to be expanded by the generator K a b .
the matrices (T C ) A B are given by They can be expressed as In general, they are not exactly E n U -duality transformations, becauseK a b is not an E n generator. Thus, suggested by [47,52], we introduce an additional generator (t 0 ) A B ≡ −δ B A for R + [52], and express the algebra as The last term in each line appears due to the fact that the generalized frame fields E A I has a density weight 1 9−n . Then, the E n algebra can be also expressed as where Θα A and θ A are constants. If we decompose the indexα as their components are (3.62) Then, we can easily obtain the matrices (T C ) A B in the R 2 -representation as follows: where (t 0 ) B A ≡ 2 δ B A . The invariance of the bilinear form under E n × R + transformations leads to the following identity:   In this sense, the Leibniz algebra E n is an extension of the Lie algebra of the Drinfel'd double.
It is noted that there exist certain Drinfel'd doubles, which are not straightforwardly embedded into the E n algebra. When the assumption (3.66) is satisfied, the Leibniz identity Then, even under the assumption (3.66), we obtain a constraint for the structure constants of the Drinfel'd double (the Leibniz identity (3.34) also may give an additional constraint). As we discuss in section 6, in the context of the Yang-Baxter (YB) deformation, the condition (3.71) is equivalent to the requirement that the classical r-matrix is unimodular. This means that, when the classical r-matrix is non-unimodular, the Lie algebra of the corresponding Drinfel'd double cannot be embedded into the E n algebra. This may be related to the fact [66] that the YB deformation for a non-unimodular r-matrix generally produces a solution of the generalized supergravity [67,68], and the fact that the embedding of the generalized supergravity into EFT is non-trivial [69] (see section 6 for further discussion).
U -duality transformation: Let us consider a redefinition of the E n generators, where C A B is an element of the E n group. We also redefine the bilinear-form as by acting the same E n transformation in the R 2 -representation. Then, the physical subalgebra is maximally isotropic even after the redefinition, On the other hand, the E n algebra is transformed as where we have defined Even under such restriction, the allowed U -duality symmetry is much larger than the case of the PL T -duality.

Generalized frame fields
In this section, we present a systematic construction method of the generalized frame fields E A I , which is analogous to the one known in the PL T -duality. Then, by following the approach of [70], we show that the E A I indeed satisfy the desired relation, where X AB C is the structure constant F AB C of the Leibniz algebra E n .
Let us prepare a set of generators T a associated with a maximal isotropic subalgebra. As already explained, the subalgebra is a Lie algebra, and we can parameterize an element of the Lie group G as usual, e.g., g = e x a Ta . We define the left-/right-invariant 1-forms/vectors as which satisfy Then, we define the action of g −1 (x) ≡ e h(x) on T A as Since the infinitesimal transformation is an E n × R + transformation of the lower-triangular form (3.54), the matrix M A B can be generally parameterized as Then, we define the generalized frame fields as where the matrix L A I is defined by and we have used ℓ a i = a b a r b i in the second equality of (4.6). This matrix E A I plays the role of the desired generalized frame fields as we show below. For this purpose, let us find several identities by following [70].
Differential identities: Differentiating the definition (4.4) of the matrix M A B , we obtain The left-hand side can be evaluated as and (4.8) gives the following identities: Algebraic identities: In order find further relations, we consider the identity, which follows from the Leibniz identity. For convenience, we decompose this identity as On the other hand, the component 14) and the component 17) and they are all identities coming from (4.11).
Computation of X AB C : By using the differential and algebraic identities, we can easily requires a slightly longer computation. The former requires the identity (4.17) while the latter requires (4.16). In this way, we have shown the desired relation X AB C = F AB C .
In summary, by using the E n algebra, we have explained a systematic construction of the generalized frame fields E A I , which satisfy the algebra of E n by means of the generalized Lie derivative. The construction is a straightforward extension of the procedure known in the PL T -duality, and we expect that this extension plays an important role in formulating the U -duality extension of the PL T -duality. When n = 2, we obtain a three-dimensional algebra with generators {T A } = {T 1 , T 2 , T 12 } .
By denoting f 12 1 = a and f 12 2 = b, we obtain The non-vanishing components of the bilinear form are This is not an interesting example, but it is a good example to clearly see the existence of another maximal isotropic subalgebra. As we can clearly see from (5.3), the generator T 12 has non-vanishing inner products with other generators. This shows that the Abelian algebra generated by {T a } = {T 12 } is another maximal isotropic subalgebra. Similarly, E n always has two types of maximal isotropic subalgebras with dimension n and n − 1 .

6D algebra E 3
The algebra E 3 is a six-dimensional algebra with generators The structure constants f ab c have 9 components and f a bcd have 3 components. According to the Bianchi classification, the 3D Lie algebra f ab c has been classified. It is interesting to classify the additional structure constants f a bcd for each physical 3D Lie algebra.

10D algebra E 4
M-theory frame I: In n = 4, the algebra E 4 is ten-dimensional and the structure is much richer. As a particular example, we here consider the case, On the other hand, the tensor Π i 1 i 2 i 3 ≡ e i 1 a 1 e i 2 a 2 e i 3 a 3 Π a 1 a 2 a 3 has the form By construction, the R-flux X a 1 a 2 b 1 b 2 c should vanish, and it satisfies In order for this to be a Nambu-Poisson tensor, the algebraic or quadratic identity, should be satisfied [64] (see also [71]). In this example, it is indeed satisfied and the above Π is a Nambu-Poisson tensor. In general, we have not checked the quadratic identity, but it may follow from a certain requirement such as the Leibniz identity.
By using the trivial right-invariant vector and the Nambu-Poisson structure, the generalized frame fields become As we have generally proven, this satisfies the relation£ E A E B I = −F AB C E C I for the structure constants given in (5.4).
M-theory frame II: Let us consider a redefinition, This is a map M-theory corresponding to a double Abelian T -duality, and it is a particular U -duality transformation.
After the redefinition, we find the new physical generators satisfy The component f ′ abc 1 c 2 is not allowed in the E 4 algebra, and we can consider this U -duality transformation only when a = 0 . 7 Moreover, the algebra of other generators further requires c = 0 . 8 Under a = c = 0, the U -duality converts the structure constants of the E 4 algebra as Again, we can easily construct the generalized frame fields realizing this algebra.
Type IIB frame: Let us consider another redefinition of the E 4 generators, This map has been considered in [72], which connects the M-theory picture and the type IIB picture (see [41,58] for earlier discussion). This is not a U -duality transformation but rather corresponds to a change in the picture, from M-theory to type IIB theory. 9 In type IIB theory, we can decompose the R 1 -representation (for n ≤ 5) as and we understand the above redefinition as This algebra should be regarded as the physical subalgebra in type IIB theory. Although the entire algebra in the type IIB picture has not been established, it seems that this example does not contain any dual structure constants, which may have the form, f a (αβ) or f a If we restrict to the case a = 0, b = −1, c = 1, and d = 0 , the algebra is Bianchi type 6 0 , In this case, by using a supergravity solution obtained in [60] (which has the symmetry of the Bianchi type 6 0 ), we can perform a U -duality extension of the PL T -duality. Namely, in 7 The flux f1 234 = a corresponds to the Q-flux Q1 23 = −a in type IIA theory and the double T -duality transforms it to the H-flux H123 = −a . The 11D uplift corresponds to f ′ 1234 = −a . 8 The reason may be understood as follows. Originally, the Π a 1 a 2 a 3 has the x 2 -dependence, but under the double T -dualities, x 2 is mapped to y24 . The dependence on the dual coordinate breaks our assumption (3.2).
Accordingly, the resulting algebra has a different form from E4 . 9 One can see that the bilinear form is not invariant under the transformation (see [72] for the transformation rule of the index A under this redefinition).
the M-theory picture, we can construct a solution of EFT that is twisted by the matrix (5.9) with a = 0, b = −1, c = 1, and d = 0 . Under the change of the generators, the solution is mapped to the type IIB solution of [60]. However, it is not so interesting because it is nothing more than the straightforward 11D uplift of the PL T -duality. In the type IIA picture, (5.4) reduces to the Lie algebra of the Drinfel'd double [recall (3.68) and choose z = 4] 20) where the physical algebra is Abelian and the dual algebra is Bianchi type 6 0 . Then, the redefinition (5.15) corresponds to a non-Abelian T -duality. In order to find genuinely Uduality examples, it is important to study the detailed classification of the E n algebra.

Summary and Discussion
Summary: When we perform the PL T -duality, a systematic construction of the generalized Straightforward extensions: In this paper, we have concentrated on the case n ≤ 4 , but the extension to higher n will be straightforward. Since the generators T A are transforming in the R 1 -representation, for higher n, we introduce the following generators: {T A } = {T a , T a 1 a 2 , T a 1 ···a 5 , T a 1 ···a 7 ,a , · · · } . (6.1) According to the success of the E 11 conjecture [35,40], it will be possible to extend n up to n = 11 . Here we have almost restricted to the M-theory picture, but if we consider the type IIB picture, the generators are parameterized as The invariant bilinear forms in both the M-theory/type IIB pictures are also well studied in EFT (see [61] for n ≤ 7). The algebra should always have the form and what we need to do for higher n will be to consistently find the constants Θα A and θ A .
The construction method of E A I also will be straightforwardly extended to higher n .
Towards non-Abelian U -duality: The most interesting application of our result is the Uduality extensions of the PL T -duality (which may be called the Nambu-Leibniz U -duality). In order to study non-trivial examples of such U -duality, the decompositions of the E n algebra into the physical and the dual subalgebras need to be classified. In the case of the Drinfel'd double, such decomposition is known as the Manin triple, and the classification for six-dimensional case has been worked out in [73]. The extension of such classification for each E n algebra is important. A major difference from the case of the Drinfel'd double is in the existence of the two types of subalgebras with dimension n and n − 1 . Another difference is that the dual algebra of E n (generated by T a 1 a 2 ) is not maximally isotropic and accordingly is a Leibniz algebra. Namely, unlike the case of the Drinfel'd double, the E n algebra is decomposed into an n-dimensional physical Lie algebra and a (D n − n)-dimensional dual Leibniz algebra. It is also noted that the E n algebra in the M-theory picture and the type IIB picture may not be exactly the same in general. In the M-theory picture, we introduced the structure constants f ca b and f c a 1 a 2 a 3 corresponding to the E n generators K a b and R a 1 a 2 a 3 but do not introduce f ca 1 a 2 a 3 that corresponds to R a 1 a 2 a 3 . On the other hand, in the type IIB picture, we may introduce corresponding to the E n generators K a b , R (αβ) , and R α a 1 a 2 , but will not introduce f c α [a 1 a 2 ] that is associated with R a 1 a 2 α . Then, the number of the structure constants does not match between the two pictures. It may coincide after imposing the Leibniz identity but it is not obvious and is important to study the correspondence in detail.
It is also important to study the flux-formulation of EFT. In the case of gauged DFT [26][27][28][29][30][31], the action and equations of motion are expressed purely by using the generalized flux F ABC (and additional flux F A ). Moreover, when the flux is constant, the equations of motion reduce to the algebraic equations (2.18). A similar analysis has been done in [52], and the action of EFT is expressed by the generalized fluxes X AB C . If the equations of motion are also expressed by using the fluxes, and if they reduce to simple algebraic equations when X AB C is constant, we can clearly see the symmetry of the non-Abelian U -duality.
Duality in the membrane sigma model: It is important to study the duality symmetry also in the context of the membrane sigma model. Originally, the PL T -dualizability condition has been found in the form, where E mn ≡ g mn − B mn . By solving the differential equation with the help of the Drinfel'd double, the twist matrix (2.9) has been obtained. The condition (6.4) shows that the equations of motion of the string sigma model are expressed as a Maurer-Cartan equation, 5) and this plays an important role in realizing the PL T -duality as a symmetry in the equation of motion of string theory. When the matrix E mn is invertible, (6.4) is equivalent to and if we define a dual metricg mn and a bi-vector β mn through the relatioñ g mn + β mn = (g + B) −1 mn , (6.7) the dualizability condition is expressed as In fact, we can easily find a similar relation in our setup. If we define the generalized metric as AB by using the twist matrix (4.6) and a diagonal constant metriĉ M IJ made of an invariant metric κ ab of the physical subgroup G, we find that the dual metric g ij (i.e. the M-theory uplift ofg mn ) and the Ω-field (i.e. the M-theory uplift of the β-field, Ω mnz = β mn ) are given bỹ Then, we can show the following relation, which is the M-theory uplift of (6.8): It is interesting to study the implication of these relations in the context of the membrane sigma model. Perhaps, the equations of motion of the membrane sigma model can be expressed in a similar form as (6.5) and it may help to discuss the non-Abelian U -duality in the context of the membrane sigma model.
Generalized Yang-Baxter deformation: Another related direction is a generalization of the YB deformation [74][75][76][77][78]. As it has been observed in [79], the YB deformation is a coordinate-dependent β-deformation β mn → β ′mn = β mn + r mn , associated with a bi-Killing vector r mn ≡ r ab v m a v n b , where r ab = −r ba is a constant matrix. Here, the set of vector fields v m a satisfies the algebra [v a , v b ] m = f ab c v m c and the Killing equation £ va (g + B) mn = 0 in the undeformed background. In this case, the YB-deformed background satisfies the PL T -dualizability condition (6.8) with the dual structure constant given bỹ Interestingly, when the matrix r ab satisfies the homogeneous classical YB equations (CYBE), the YB deformation always maps a DFT solution to another DFT solution. The reason can be clearly understood by noticing that the YB deformation is a specitic PL T -duality.
Before the YB deformation, the background satisfies £ va (g + B) mn = 0 and this shows thatf ab c = 0 . Namely, in the original background, which is described by the generalized where κ ab and β ab are constant, and κ ab is supposed to be an invariant metric of the isometry algebra. The YB deformation corresponds to the PL T -duality (2.19) with Under this transformation, the generators become T ′ a = T a and T ′a = T a − r ab T b and the constant fields are transformed as κ ab → κ ′ ab = κ ab and β ab → β ′ab = β ab + r ab . By requiring that the new generators T ′ A satisfy the Lie algebra of the Drinfel'd double withf b 1 b 2 a given by (6.11), the matrix r ab must be a classical r-matrix satisfying (6.12). Then, using the systematic construction of E ′ A M , we can in principle compute the generalized frame fields 15) and the β-field in the deformed background can be computed as β ′mn = β mn + π mn , π mn ≡ e m a e n b Π ab + r ab . (6.16) The great benefit of the YB deformation is that we do not need to compute π mn . It is simply given by π mn = r mn because r mn solves the differential equation £ va r mn = −f bc a v m b v n c , and π mn = r mn is trivially satisfied at the identity g = 1 [recall (2.15)]. Thus, once we find a classical r-matrix, we can easily generate a new solution. In this sense, the YB deformation is a systematic way to perform the PL T -duality (6.14) and the homogeneous CYBE ensure that the structure of the Drinfel'd double is preserved under the deformation.
Recently, an 11D extension of this YB deformation has been studied in [80]. There, the YB deformation is generalized to the Ω-deformation Ω i 1 i 2 i 3 → Ω ′i 1 i 2 i 3 = Ω i 1 i 2 i 3 + ρ i 1 i 2 i 3 associated with a tri-Killing vector where ρ a 1 a 2 a 3 = ρ [a 1 a 2 a 3 ] is a certain constant. By assuming the Killing equations (£ va g ij = 0 and £ va C i 1 i 2 i 3 = 0) in the undeformed background, the Ω-deformed background satisfies the relation (6.10) with the dual structure constant given by Similar to the case of the YB deformation, this also can be understood as a specific non-Abelian U -duality transformation (3.72) with By requiring that the redefined generators T ′ a = T a and T ′a 1 a 2 = T a 1 a 2 + ρ a 1 a 2 b T b to satisfy the E n algebra, we obtain The last requirement is a natural generalization of the homogeneous CYBE and the former two equations are intrinsic to the E n algebra [which correspond to (4.15) and (4. 16)]. Again, the Ω-field after the deformation is given by The tri-Killing deformation can be understood as the M-theory uplift of the YB deformation in type IIA theory, where the parameter ρ a 1 a 2 a 3 is related to the r-matrix as ρȧ˙b z = rȧ˙b .
Then, the first equation in (6.20) is reduced to the unimodularity condition f˙b 1ḃ2ȧ r˙b 1ḃ2 = 0 . 10 This unimodularity condition is precisely the condition for the YB-deformed background to satisfy the supergravity equations of motion [66]. However, the Lie algebra of the Drinfel'd double itself is consistently defined even when the unimodularity is violated. Moreover, as it is shown in [66], even in the non-unimodular case, the YB-deformed background does satisfy the equations of motion of the generalized supergravity [67,68], and it is also a solution of DFT [81]. Then, a natural question is why the non-unimodular cases are excluded from the tri-vector deformation based on the E n algebra. This can be understood as follows.
In the case of non-unimodular YB deformations, the deformed geometries are solutions of the generalized supergravity, which means that the dilaton in type IIA theory acquires a dependence on the dual coordinatesx m = y mz [82]. In the case of DFT, the dilaton is not 10 The Leibniz identity (3.71) for the dual structure constant (6.11) also reproduces the same condition. contained in the generalized frame fields E A M and this does not cause any problem in realizing the algebra of the Drinfel'd double as£ E A E B M = −F AB C E C M . However, in EFT, the dilaton is contained in the generalized frame fields E A I and the dual-coordinate dependence conflicts with our assumption (3.2). This will be the reason why the Drinfel'd double associated with non-unimodular YB deformation cannot be embedded into the E n algebra. In order to study the non-unimodular YB deformation in the context of EFT, it may be necessary to deform the E n algebra by changing the choice of the section (3.2). Such deformation of the E n algebra may be realized also by considering a deformation of the generalized Lie derivative as it has been considered in [83,84], because the introduction of the dual-coordinate dependence is equivalent to the introduction of the deformation parameters.
Connection with mathematics: It is interesting to investigate connections with various known facts in the mathematical literature. As we have mentioned, the E n algebra is related to the Nambu-Poisson tensor, and various results on the Nambu-Poisson group (see for example [65,71,85]) will be useful to clarify the structure of the E n algebra. In addition, the E n algebra is a Leibniz algebra (rather than a Lie algebra), and it seems to have an intricate global structure. The detailed study of such global structure is also an interesting future direction.