aa r X i v : . [ h e p - t h ] S e p Extended Drinfel’d algebras and non-Abelian duality
Yuho Sakatani
Department of Physics, Kyoto Prefectural University of Medicine,Kyoto 606-0823, Japan [email protected]
Abstract
A Drinfel’d algebra gives the systematic construction of generalized parallelizable spacesand this allows us to study an extended T -duality, known as the Poisson–Lie T -duality.Recently, in order to find a generalized U -duality, an extended Drinfel’d algebra (ExDA),called the Exceptional Drinfel’d algebra (EDA) was proposed and a natural extension ofthe usual U -duality was studied both in the context of supergravity and membrane theory.In this paper, we clarify the general structure of ExDAs and show that an ExDA alwaysgives a generalized parallelizable space, which may be regarded as a group manifold withgeneralized Nambu–Lie structures. We also discuss generalized Yang–Baxter deformationsthat are based on coboundary ExDAs. As important examples, we consider the E n ( n ) EDAfor n ≤ ontents R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Y -tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 More on the generalized Lie derivative . . . . . . . . . . . . . . . . . . . . . . . 12 E AI . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1.3 Structure constants of ExDA . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Leibniz identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.1 Cocycle conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.2 Fundamental identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Generalized frame fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3.1 Important identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4 Generalized parallelizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4.1 Nambu–Lie structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.5 Generalized classical Yang–Baxter equation . . . . . . . . . . . . . . . . . . . . 28 A.1 E n ( n ) algebra in the M-theory picture . . . . . . . . . . . . . . . . . . . . . . . 48A.2 E n ( n ) algebra in the type IIB picture . . . . . . . . . . . . . . . . . . . . . . . . 50A.3 Explicit matrix form of χ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 Introduction
The Poisson–Lie T -duality [1, 2] is an extension of the well-established Abelian T -duality.This is based on a Lie group, called the Drinfel’d double. The Drinfel’d double contains a(maximally isotropic) subalgebra g and the group manifold of G = exp g plays the role of thetarget space of string theory. A Drinfel’d double generally contains various subalgebras g ,and a different choice of the subalgebra yields a different target space. Since the equations ofmotion of supergravity or string theory are insensitive to the choice of subalgebra, arbitrarinessin the choice of the subalgebra gives rise to the non-trivial target space duality.String theory has a larger duality group called U -duality, and a natural question is whetherthere are U -duality analogue of the Poisson–Lie T -duality. In order to address this question, anextended Drinfel’d algebra was proposed in [3,4]. Since the U -duality group is the exceptionalgroup E n ( n ) , this algebra is called the exceptional Drinfel’d algebra (EDA). Similar to thecase of the Poisson–Lie T -duality, the EDA contains various maximally isotropic subalgebrasand each subalgebra gives a target space. Whether this extended U -duality can be used asa solution generating technique in supergravity is still an open question, but the answer ispositive as discussed in [3–5] by using several examples. In addition, as was discussed in [6],the equations of motion of the (topological) membrane worldvolume theory can be expressedin a covariant way under the extended U -duality. Encouraged by these results, further detailsof the EDA have been studied recently (see [7–9] for recent works).In the original works [3, 4], the E n ( n ) EDA was proposed for n ≤ n ≤ n increases thedimension of the EDA and makes it more difficult to study various properties of the EDA.The main purpose of this paper is to study the properties of the E n ( n ) EDA in a unified waythat is less dependent on n . In fact, most of the properties of the EDA do not depend on thedetails of the duality group G , and thus we consider an extended Drinfel’d algebra (ExDA)associated with a general group G . When G is the O( d, d ) T -duality group, the ExDA reducesto the Lie algebra of the Drinfel’d double while when G is the E n ( n ) U -duality group, theExDA reduces to the E n ( n ) EDA. In general, other duality groups are also possible, but inthis paper, we will focus on these two cases.We also present the E n ( n ) EDA for n ≤
8, as an application of the general discussion of theExDA. Previous studies of the EDA have adopted the M-theory picture, where the dimensionof the subalgebra g is n , but as discussed in a series of studies [10–13], U -duality covarianttensors can be decomposed into tensors in both M-theory and type IIB theory. In this paper,in addition to the M-theory picture, we also construct the EDA in type IIB picture. In thetype IIB picture, the dimension of g is n − .1 A brief sketch of Non-Abelian Duality In order to illustrate the main results, we here explain the Poisson–Lie T -duality in moredetail. For this purpose, it is very useful to use the language of the Extended Field Theory(ExFT) which includes Double Field Theory (DFT) [14–18] and Exceptional Field Theory(EFT) [19–26] as special cases, G = O( d, d ) and G = E n ( n ) , respectively. ExFT is a manifestly G -covariant formulation of supergravity, extending the physical space of n dimensions intoan extended space of D dimensions. The details are reviewed in section 3, but of particularimportance is that the infinitesimal diffeomorphisms are generated by the generalized Liederivative ˆ £ V , which is a modification of the usual Lie derivative £ v . In the following, weexplain that the existence of certain frame fields E AI ( A, I = 1 , . . . , D ) in DFT that satisfyˆ £ E A E BI = − X ABC E C I , (1.1)where X ABC are the structure constants of the Drinfel’d algebra, is very important in thePoisson–Lie T -duality (see [27] where this was pointed out).Let us consider a string sigma model, known as the double sigma model [28–30], S = − Z d σ (cid:2) M IJ ( x ) ∂ x I ∂ x J − η IJ ∂ x I ∂ x J (cid:3) , (1.2)where 2 πα ′ = 1 and I, J = 1 , . . . , D ( ≡ d ) . We decompose the index as ( x I ) = ( x m , ˜ x m )( m = 1 , . . . , d ) and parameterize the generalized metric M IJ and the O( d, d ) metric η IJ as M IJ ≡ g mn − B mp g pq B qn B mp g pn − g mp B pn g mn , η IJ = δ nm δ mn . (1.3)By assuming that M IJ depends only on x m , the equations of motion reproduce the standardequations of motion for x m ( σ ) , and the double sigma model is (classically) equivalent to the(bosonic) string theory (see [30] for details). In particular, when the target space is constant,the action is manifestly invariant under the O( d, d ) T -duality transformation x I → (Λ − ) J I x J , M IJ → Λ I K Λ J L M KL (Λ ∈ O( d, d )) . (1.4)In the canonical formalism, we define the conjugate momenta as P I = η IJ ∂ x J , (1.5)and define the Poisson bracket as { x I ( σ ) , P J ( σ ′ ) } = 2 δ IJ δ ( σ − σ ′ ) , { x I ( σ ) , x J ( σ ′ ) } = 0 = { P I ( σ ) , P J ( σ ′ ) } . (1.6)The Hamiltonian is obtained as H = 12 Z d σ M IJ ( x ) ∂ x I ∂ x J . (1.7)3e find that there are the second-class constraints e P I ≡ P I − η IJ ∂ x J , and the Dirac bracketis obtained as[ x I ( σ ) , x J ( σ ′ )] = − η IJ ǫ ( σ − σ ′ ) , [ x I ( σ ) , P J ( σ ′ )] = δ IJ δ ( σ − σ ′ ) , [ P I ( σ ) , P J ( σ ′ )] = η IJ δ ′ ( σ − σ ′ ) . (1.8)Then the dynamics is described by this Dirac bracket and the Hamiltonian (1.7).Now we consider the case where the generalized metric is of the form M IJ ( x ) = E I A ( x ) E J B ( x ) ˆ H AB , (1.9)where ˆ H AB is constant and E I A ( x ) is the inverse matrix of the generalized frame fields E AI ( x )obeying Eq. (1.1). If we define the currents j A ( σ ) ≡ E AI ( x ( σ )) P I ( σ ) , (1.10)the Hamiltonian can be expressed as H = 12 Z d σ ˆ H AB j A ( σ ) j B ( σ ) . (1.11)Using Eq. (1.1) and the Dirac bracket (1.8), we obtain[ j A ( σ ) , j B ( σ ′ )] = X ABC j C ( σ ) δ ( σ − σ ′ ) + η AB δ ′ ( σ − σ ′ ) , (1.12)and the physical system governed by this current algebra and the Hamiltonian (1.11) is calledthe E -model [31–33]. Under this setup, we can discuss the Poisson–Lie T -duality as follows.Suppose that there exists another set of generalized frame fields E ′ AI that satisfyˆ £ E ′ A E ′ BI = − X ′ ABC E ′ C , X ′ ABC ≡ C AA ′ C BB ′ ( C − ) C ′ C X A ′ B ′ C ′ , (1.13)where C AB ∈ O( d, d ) is a constant matrix and E ′ AI = C AB E BI . Then we can define anothercurrents j ′ A ( σ ) ≡ E ′ AI ( X ( σ )) P I ( σ ) obeying[ j ′ A ( σ ) , j ′ B ( σ ′ )] = X ′ ABC j ′ C ( σ ) δ ( σ − σ ′ ) + η AB δ ′ ( σ − σ ′ ) . (1.14)Introducing a new Hamiltonian H ′ = 12 Z d σ ˆ H ′ AB j ′ A ( σ ) j ′ B ( σ ) , ˆ H ′ AB ≡ C AC C BD ˆ H CD , (1.15)we see that the dynamics in the primed system is the same as the original one because the roleof j A is played by C AB j ′ B . This shows that string theory defined on the original background(1.9) is equivalent to that defined on another curved background H ′ IJ = E ′ I A ( X ) E ′ J B ( X ) ˆ H ′ AB (cid:0) = H IJ (cid:1) . (1.16)4his is the Poisson–Lie T -duality. Here, whether we can actually find the generalized framefields E AI and E ′ AI satisfying (1.1) and (1.13) is very important. As was found in [1, 2] (andin [27] in the language of DFT), by using the Drinfel’d algebra, we can systematically constructsuch generalized frame fields E AI and E ′ AI . The systematic construction of the generalizedframe fields is the great benefit of the Drinfel’d algebra.The same discussion holds also for membrane theory [6], where the T -duality group isextended to the U -duality group. In membrane theory, the extended momenta P I ( σ ) satisfythe Poisson bracket [34] [ P I ( σ ) , P J ( σ ′ )] = ρ a IJ ( σ ) ∂ a δ ( σ − σ ′ ) , (1.17)where σ a = 1 , ρ a IJ ( σ ) is a certain non-constantmatrix. The Hamiltonian can be expressed as H = 12 Z d σ √ h M IJ ( x ) P I P J , (1.18)similar to the double sigma model. Again, we suppose that the generalized metric has theform (1.9) and assume that the generalized frame fields satisfiesˆ £ E A E BI = − X ABC E C , (1.19)by means of the generalized Lie derivative in EFT. Here, X ABC are the structure constants ofthe EDA and, in general, the lower indices are not antisymmetric X ABC = X [ AB ] C (namely,the EDA is a Leibniz algebra). Then, by defining the currents j A ( σ ) ≡ E AI ( X ( σ )) P I ( σ ), theHamiltonian and the Poisson bracket of the currents become H = 12 Z d σ √ h ˆ H AB j A ( σ ) j B ( σ ) , [ j A ( σ ) , j B ( σ ′ )] = X [ AB ] C j C ( σ ) δ ( σ − σ ′ ) + 12 (cid:2) ρ a AB ( σ ) − ρ a AB ( σ ′ ) (cid:3) ∂ a δ ( σ − σ ′ ) . (1.20)We thus obtain a natural extension of the E -model [6]. At least when the target space is threedimensional, the membrane theory has the E U -duality covariance. Then, similar to thecase of T -duality, we can consider E ′ AI satisfying (1.13) and we find that membrane theoryhas the symmetry under the non-Abelian U -duality [6].As described above, the existence of the generalized frame fields E AI satisfying Eq. (1.19) isvery important in realizing the non-Abelian duality. When such generalized frame fields exist,the target space is called a generalized parallelizable space [36–38] and that plays an importantrole in considering a dimensional reduction preserving supersymmetry. As we show in thispaper, a major advantage of the ExDA is that, once an ExDA and its maximally isotropicsubalgebra are specified, a generalized parallelizable space can be explicitly constructed. When the dimension of the target space is greater than three, the rotation P I → Λ I J P J (Λ IJ ∈ E n ( n ) ) isnot a symmetry because it breaks some integrability [35], and even the Abelian U -duality cannot be realized. .2 Summary of results Now that we have explained the importance of ExDA, we turn to the details of the ExDA andexplain the main results. To discuss the general properties of the ExDA for a general dualitygroup G , we shall use a notation of the embedding tensor formalism developed in gaugedsupergravity [39–48]. We consider a D -dimensional Leibniz algebra T A ◦ T B = X ABC T C , (1.21)which has the structure constants of the form X ABC = Θ A α ( t α ) BC − (cid:2) β ( t α ) AD ( t α ) BC + δ DA δ CB (cid:3) ϑ D , Θ A α ≡ P A α B β F B β , ϑ A ≡ F A − β F B α ( t α ) AB . (1.22)Here, { t α } are the generators of the duality group G , β is a constant specific to G , P isa projection on a certain representation, and the information of the structure constants iscontained in F A ≡ F A α t α + F A t ( t : generator of a scale symmetry R + [49, 50]).So far, this is the general structure of the gauge algebra for maximal gauged supergravities(in the presence of the trombone gauging) [48]. The additional requirement specific to theExDA is that this Leibniz algebra contains an n -dimensional subalgebra g , [ T a , T b ] = f abc T c ,that is maximally isotropic with respect to some bilinear form h T a , T b i = 0 . This requirementand the consistency condition of ExFT imply that F A can be parameterized as F A = δ bA (cid:0) F bcd K cd + F b ˆ A R ˆ A + F a t (cid:1) , (1.23)where K ab is the generator of the gl ( n ) subalgebra and { K ab , R ˆ A } is a set of non-positive-levelgenerators of G . Substituting this F A into Eq. (1.22), we obtain an ExDA.If an ExDA and its maximally isotropic subalgebra g is specified, we can construct thegeneralized frame fields E AI by following the procedure known in the Poisson–Lie T -duality.In the previous studies, it was very hard to check that the constructed E AI indeed satisfythe relation (1.19). However, thanks to the symbolic expression (1.22), here we can generallyshow the relation for a wide class of duality groups G . Namely, we show that the ExDA alwaysconstructs a generalized parallelizable space for an arbitrary choice of the subalgebra g . Thisis one of the main results and plays an important role in the non-Abelian duality.As known in the Drinfel’d algebra or the E n ( n ) EDA ( n ≤ r -matrix. We propose the generalized classical Yang–Baxter equationby extending the results of [3, 4, 9]. We also discuss the Yang–Baxter deformation, which is aspecific example of the non-Abelian duality based on the classical r -matrices.6his paper is organized as follows. In section 2, we briefly review the necessary ingredientsof ExFT. There, several non-standard notation for the duality algebra is also introduced. Insection 3, we introduce the ExDA and explain various general properties. In section 4, weexplain how to generate solutions of ExFT through the non-Abelian duality. In sections 5 and6, we study the E n ( n ) EDA in the M-theory picture and the type IIB picture, respectively.Section 7 is devoted to summary and discussions.
In this section, we set up our notation by shortly reviewing ExFT (see [51] for a review).The well-studied ExFTs are DFT and EFT where the duality group G is O( d, d ) and E n ( n ) ,respectively. In this paper, we particularly focus on E n ( n ) EFT with n ≤ G , as much as possible.In ExFT, we introduce a D -dimensional extended space with the generalized coordinates x I ( I = 1 , . . . , D ). Diffeomorphisms in the extended space is generated by the generalized Liederivative (see [22] for details)ˆ £ V W I ≡ V J ∂ J W I − W J ∂ J V I + Y IKJL ∂ K V L W J , (2.1)where Y IJKL is some invariant tensor that is explained in section 2.3. For consistency, for allfields defined on the extended space, we impose the so-called section condition Y IJKL ∂ I ⊗ ∂ J = 0 . (2.2)According to this condition, all fields are defined on a maximally isotropic subspace (calledthe physical subspace) with coordinates x i ( i = 1 , . . . , n ). After choosing such a subspace,representations of the duality group can be decomposed into representations of the GL( n )subgroup, which is associated with coordinates transformations on the physical space. The set of generators { t α } of each duality group G can be decomposed as follows:O( d, d ) : { t α } = { R a a √ , K ab , R a a √ } ( a, b = 1 , . . . , d ) , (2.3) E n ( n ) : { t α } = { R a ··· a ,a ′ √ , R a ··· a √ , R a a a √ , K ab , R a a a √ , R a ··· a √ , R a ··· a ,a ′ √ } (M-theory) ( a, b = 1 , . . . , n = n ) , (2.4)where the multiple indices are totally antisymmetric. For G = E n ( n ) , this decomposition issuitable for M-theory, but there exists another maximally isotropic subspace with n = n − n − × SL(2): E n ( n ) : { t α } = (cid:8) R a ··· a , a ′ √ , R α a ··· a √ , R a ··· a √ , R α a a √ , K ab , R αβ , R a a α √ , R a ··· a √ , R a ··· a α √ , R a ··· a , a ′ √ (cid:9) (Type IIB) ( a , b = 1 , . . . , n = n − , α = 1 , . (2.5)Here, R αβ ( R αα = 0) are the generators of the SL(2) , which is associated with the S -dualitysymmetry of type IIB theory.For generators in each decomposition, we introduce an ordering, called the level ℓ α , as[ K, t α ] = σ ℓ α t α ( K ≡ K aa ) . (2.6)Here, the normalization constant σ is given by σ = 2, 3, 2 for O( d, d ), E n ( n ) (M-theory), and E n ( n ) (type IIB), respectively. This σ is introduced such that the level coincides with the valueknown in the literature [52], but it is not important in our discussion. Explicitly, the level foreach generator is given as follows: O( d, d ) level − t α R K R , (2.7)M-theory level − − − t α R , R R K R R R , , (2.8)Type IIB level − − − − t α R , R α R R α K , R αβ R α R R α R , . (2.9)Here we have denoted the GL( n ) indices schematically, for example, as R , ≡ R a ··· a ,a ′ . TheJacobi identity shows that the duality algebra[ t α , t β ] = f αβγ t γ , (2.10)respects the level, in the sense that the commutator of a level- p generator and a level- q generator is a combination of level-( p + q ) generators.For later convenience, we denote the set of positive-level generators as { R A } and thenegative-level generators as { R A } . We also denote the set of level-0 generators other than K ab as { R I } . Then, for each duality group, we haveO( d, d ): { R A } = { R } , { R A } = { R } , { R I } = ∅ , (2.11)M-theory: { R A } = { R , R , R , } , { R A } = { R , R , R , } , { R I } = ∅ , (2.12)Type IIB: { R A } = { R α , R , R α , R , } , { R A } = { R α , R , R α , R , } , { R I } = { R αβ } . (2.13) We use the indices a , b in the type IIB picture, but also use a, b when no confusion arises. By the definition, σ ℓ α coincides with (number of upper GL indices) − (number of lower GL indices).
8e note that the commutation relations among the E n ( n ) generators both in the M-theoryand the type IIB picture are given in Appendix A. The commutation relations for the O( d, d )algebra are given in Eq. (3.30). In each duality group, we see that the generators are repre-sentation of the subalgebra h generated by the level-0 generators { K ab , R I } .The properties of the duality group G that are important in the following discussion canbe summarized as follows:1. The generators { t α } can be decomposed into the positive-level generators { R A } , negative-level generators { R A } , and the level-0 generators { K ab , R I } .2. { R A } and { R A } are representations of the subalgebra h generated by { K ab , R I } . R The usual momenta P a are related to various brane central charges via duality. For example,under T -duality, the momenta P a are mapped to the string winding charges P a , and under U -duality, they are mapped to brane central charges. All of the central charges are packagedto the generalized momenta P A and they transform in the vector representation of G whichwe call the R representation. In each duality group, we can summarize the contents of thegeneralized momenta as follows: O( d, d ): { P A } = { P , P } , (2.14)M-theory: { P A } = { P , P , P , P , , P , , P , , P , , } , (2.15)Type IIB: { P A } = { P , P α , P , P α , P , , P , α , P , , P , α , P , , } . (2.16)Since the generalized momenta are in a representation of the duality group, we have[ t α , P A ] = ( t α ) AB P B . (2.17)By assuming the central charges are Abelian [ P A , P B ] = 0 and using the Jacobi identities, theexplicit forms of the matrices ( t α ) AB were determined in [12, 20, 53, 54] (see also [55]). Theyprovide the matrix representation of the generators { t α } in the R representation.We can also assign the level to the central charges P A as[ K, P A ] = (cid:0) σ ℓ A − (cid:1) P A . (2.18)For example, P a has level 0 and other central charges have positive levels. Then, by consideringthe level, we find important properties( R A ) aB = 0 , ( R A ) Ab = 0 , ( R I ) Ab = 0 . (2.19) For G = E , P , in M-theory and P , in type IIB theory are reducible. For example, P , canbe decomposed into P a ··· a ,a ′ with P [ a ··· a ,a ′ ] = 0 and the totally antisymmetric part P . However, thisdecomposition increase the matrix size of ( t α ) AB and we do not consider such a decomposition. R A , P a ] = ( R A ) aB P B = (sum of the central charges with negative level) = 0 , (2.20)and the second can be also understood in a similar way. The last property follows fromthe fact that P a is a singlet under the subalgebra generated by { R I } : [ P a , R I ] = 0 . Theseproperties are very important in the construction of ExDA, and in the following discussionwe frequently use them implicitly. Moreover, since each component of the central charges arerepresentations of h , the matrices ( K cd ) AB and ( R I ) AB have the block-diagonal form, whichis also used frequently. Y -tensor Now we explain the definition of the Y -tensor Y IJKL . This is defined as [22] Y IJKL ≡ δ IK δ JL + ( t α ) K J ( t α ) LI + β δ IL δ JK , (2.21)where β is given by G = O( d, d ) : β = 0 , G = E n ( n ) : β = 19 − n , (2.22)and { t α } is defined as follows. For a while, we suppose that the duality group G is simple,and define the Cartan–Killing form as κ αβ ≡ − α tr R ( t α t β ) , ( κ αβ ) ≡ ( κ αβ ) − , (2.23)where the constant α for each duality group is G O( d, d ) E E E E E α . (2.24)Then we define t α as t α ≡ κ αβ t β . (2.25)In our convention, the are found as follows:O( n, n ): { t α } = { R , K , R } , M-theory: { t α } = { R , , R , R , K , R , R , R , } , Type IIB: { t α } = { R , , R α , R , R α , K , R αβ , R α , R , R α , R , } , (2.26)where K and R αβ are defined asO( n, n ): K ab ≡ − K ba , M-theory: K ab ≡ − (cid:0) K ba − δ ba K (cid:1) , Type IIB: K ab ≡ − (cid:0) K ba − δ ba K (cid:1) , R αβ ≡ − R βα . (2.27)10or non-simple U -duality groups E n ( n ) ( n ≤ { t α } .Then, we find the identitiesO( d, d ) : ( t α ) AB ( t α ) C D = − P adj ) ABC D , (2.28) n = 2 : ( t α ) AB ( t α ) C D = − ( P ) ABC D − ( P ) ABC D , (2.29) n = 3 : ( t α ) AB ( t α ) C D = − P (8 , ) ABC D − P (1 , ) ABC D , (2.30) n ≥ t α ) AB ( t α ) C D = − α ( P adj ) ABC D , (2.31)and ( t α ) AB ( t α ) C D is always the projector on the adjoint representation (see Table A.1). Now,the right-hand side of (2.21) is defined and the Y -tensor is explicitly defined.For example, when G = O( d, d ), we can show Y IJKL = δ IK δ JL + ( t α ) K J ( t α ) LI = η IJ η KL , (2.32)where η IJ is the inverse matrix of η IJ .For the U -duality group G = E n ( n ) , the Y -tensor can be expressed as a sum of projectorson the section-condition multiplet R SC (see Table A.1) [22] n = 2 : Y IJKL = 2 ( P ) IJ KL ,n = 3 : Y IJKL = 4 ( P (3 , ) IJ KL ,n = 4 : Y IJKL = 6 ( P ) IJ KL ,n = 5 : Y IJKL = 8 ( P ) IJ KL ,n = 6 : Y IJKL = 10 ( P ) IJ KL ,n = 7 : Y IJKL = 12 ( P ) IJ KL −
28 ( P ) IJ KL ,n = 8 : Y IJKL = 14 ( P ) IJ KL −
30 ( P ) IJ KL + 62 ( P ) IJ KL . (2.33)For each n , the projector on the largest irreducible representation in R SC has the symme-tries ( P ··· ) IJ KL = ( P ··· ) ( IJ ) KL = ( P ··· ) IJ ( KL ) . Moreover, similar to the DFT case, they areexpressed as a quadratic form of the η -symbol, η IJ K = η JI K [56, 57],2 ( n −
1) ( P ··· ) IJ KL = η IJ I η KL I . (2.34)For the second largest irreducible representation in n = 7 , P ··· ) IJ KL = ( P ··· ) [ IJ ] KL = ( P ··· ) IJ [ KL ] , (2.35)and for the smallest irreducible representation in n = 8, the indices are symmetric( P ) IJ KL = ( P ) ( IJ ) KL = ( P ) IJ ( KL ) . (2.36) The representation is called the section-condition multiplet because the section condition is expressed as(2.2) by using the Y -tensor. .4 More on the generalized Lie derivative Here we explain the generalized Lie derivative in more detail. In ExFT, all fields and gaugeparameters should satisfy the section condition (2.2). As long as the section condition issatisfied, the algebra of the generalized Lie derivative is closed, in the sense that, for givengeneralized vector fields V I , V I , V I , we can find a vector field V I satisfying[ ˆ £ V , ˆ £ V ] V I = ˆ £ V V I . (2.37)However, the algebra is not closed in E EFT [22]. In [26], by introducing additional gaugeparameter Σ I , the generalized Lie derivative was modified asˆ £ ( V, Σ) W I ≡ ˆ £ V W I + Σ K f KIJ W J , (2.38)where f KIJ corresponds to the structure constants of the e algebra. In this approach, byrequiring that Σ I satisfies certain conditions similar to the section condition, it was shownthat the modified algebra is closed in the following sense:[ ˆ £ ( V , Σ ) , ˆ £ ( V , Σ ) ] V I = ˆ £ ( V , Σ ) V I . (2.39)This approach has the advantage that it can be applied to any curved space. However, thegeometric meaning of Σ I is not clear. As explained in section 1, we would like to define thestructure constants X ABC of the ExDA through the relationˆ £ E A E BI = − X ABC E C I , (2.40)but since the meaning of Σ I is not clear, it is not obvious how to extend this relation to the E case. We thus take another approach proposed in [58]. There, by assuming the existenceof certain generalized frame fields E AI ∈ G × R + , the gauge parameter Σ I was chosen asΣ I = − α f IJ L ˜Ω KLJ V K , ˜Ω IJ K ≡ Ω I ˙ α ( t ˙ α ) J K , (2.41)where Ω is the Weizenb¨ock connectionΩ IJ K ≡ E AK ∂ I E J A ≡ Ω I ˙ α ( t ˙ α ) J K + Ω I ( t ) J K , (2.42)and ( t ˙ α ) I J is the matrix representation of the e generator in the “curved” indices. Thismodified generalized Lie derivativeˆ £ V W I ≡ V J ∂ J W I − W J ∂ J V I + Y IKJL ∂ K V L W J + Σ K f KIJ V J , (2.43)was shown to be closed in the usual sense (2.37). A disadvantage of this approach is that thisrequires the particular generalized frame fields E AI , but in the discussion of the ExDA, wehave globally defined generalized frame fields E AI . Thus we use the corresponding Weizenb¨ockconnection and adopt the generalized Lie derivative (2.43).12ince the expression (2.43) contains a quantity f IJ K that is specific to the E EFT, werewrite the definition asˆ £ V W I ≡ V J ∂ J W I − W J ∂ J V I + Y IKJL ∂ K V L W J − Σ ˙ α ( t ˙ α ) J I W J (cid:0) Σ ˙ α ≡ χ I ˙ α Ω I ˙ β χ ˙ β K V K (cid:1) , (2.44)where the matrix χ ˙ α I and its inverse χ I ˙ α are given in Appendix A.3. In the E EFT, χ is an intertwiner connecting two representations R and R adj , but it vanishes in the E n ( n ) EFT ( ≤ χ is not present. This generalized Lie derivativereproduces (2.43) by considering Σ I ≡ Σ ˙ α χ ˙ α I and f IJ K ≡ − χ I ˙ α ( t ˙ α ) K J in the E EFT,and reduces to the ordinary generalized Lie derivative in the E n ( n ) EFT ( ≤
7) and DFT.By using Eq. (2.21), it is convenient to express the generalized Lie derivative asˆ £ V W I ≡ V J ∂ J W I − (cid:2) ( ∂ × ad V ) ˙ α + Σ ˙ α (cid:3) ( t ˙ α ) J I W J − β ( ∂ · V ) ( t ) J I W J , ( V × ad W ) ˙ α ≡ − ( t ˙ α ) LK V K W L , ( V · W ) ≡ V I W I , ( t ) I J ≡ − δ JI . (2.45)This expression clearly shows that the generalized Lie derivative generates an infinitesimal G transformation and a scale transformation R + generated by t . The constant β can beunderstood as the density weight of the gauge parameter V I .For a given set of generalized frame fields E AI satisfying the section condition, we canexpand the generalized Lie derivative asˆ £ E A E BI = − X ABC E C I , (2.46)where the generalized fluxes X ABC are not constant in general. By using Eq. (2.45), thegeneralized fluxes can be expressed as X ABC = Ω
ABC + ( t α ) DE ( t α ) BC Ω EAD − β Ω DAD ( t ) BC + χ D α Ω D β χ β A ( t α ) BC , (2.47)where we have converted the “curved” indices I, J, ˙ α to the “flat” indices A, B, α by usingthe frame fields E AI and, for example, the Weizenb¨ock connection with the “flat” indices isΩ ABC ≡ E AI E BJ E K C Ω IJ K = E AI E BK ∂ I E K C . (2.48)Using the expansion of the Weizenb¨ock connection,Ω ABC = Ω A α t α + Ω A t , (2.49)we find X ABC = (cid:2) δ DA δ βα + ( t β t α ) AD + χ β A χ D α (cid:3) Ω D β ( t α ) BC − β Ω DAD ( t ) BC − ( t α ) AD Ω D ( t α ) BC + Ω A ( t ) BC . (2.50) The flat and curved adjoint indices are connected by E ˙ αα , which is defined as E AI ( t ˙ α ) IJ E JB = E ˙ αα ( t α ) AB such that the matrix elements of the ( t α ) BC and ( t ˙ α ) IJ are the same.
13y introducing ϑ A ≡ − D X ABB = (1 + β ) Ω A − β Ω D α ( t α ) AD , (2.51)we can further rewrite the generalized fluxes as X ABC = Θ A α ( t α ) BC − (cid:2) β ( t α ) AD ( t α ) BC + δ DA δ CB (cid:3) ϑ D , (2.52)where Θ A α ≡ P A α B β Ω B β , P A α B β ≡ δ BA δ αβ + ( t β t α ) AB − β β ( t α t β ) AB + χ β A χ B α . (2.53)In DFT (where β = ϑ = 0), P corresponds to the projector on the multiplet of the 3-form P A α B β = δ BA δ αβ + ( t β t α ) AB = 3 P ( d ) , (2.54)and the embedding tensor is a 3-form. Accordingly, the generalized fluxes (with the last indexlowered with η AB ) are totally antisymmetric X ABC = X [ ABC ] .In EFT with n ≤ P A α B β is always a projector on the R − n representation (see forexample the Appendix of [41] for n = 5 , , n = 2 : P A α B β = ( P ) A α B β + ( P ) A α B β ,n = 3 : P A α B β = 2 ( P (6 , ) A α B β + ( P (¯3 , ) A α B β ,n = 4 : P A α B β = 3 ( P ) A α B β + 4 ( P ) A α B β ,n = 5 : P A α B β = 4 ( P ) A α B β ,n = 6 : P A α B β = 5 ( P ) A α B β ,n = 7 : P A α B β = 7 ( P ) A α B β ,n = 8 : P A α B β = 14 ( P ) A α B β + 62 ( P ) A α B β . (2.55)Thus, the embedding tensor Θ A α in each dimension transforms in the R − n representation.Before closing this section, let us provide an additional useful information. In order toevaluate the generalized fluxes X ABC , we need the explicit form of the matrix ( t α ) AB . Inparticular, the component ( K cd ) AB frequently appears. In DFT and EFT both in the M-theory and the type IIB picture, ( K cd ) AB always has the following form: K cd ≡ − δ da δ bc ∗ , (2.56)where ∗ depends on the details of the ExFT. In [58], the modification of the generalized Lie derivative in n = 8 was motivated by this relation. General structure of ExDA
The ExDA is a Leibniz algebra with certain structure constants X ABC , T A ◦ T B = X ABC T C . (3.1)As explained in section 1.1, in order to realize the non-Abelian duality, we need a set ofgeneralized frame fields E AI that satisfies the algebraˆ £ E A E BI = − X ABC E C I . (3.2)This determine the possible forms of the structure constants X ABC . Recalling the generalform of the generalized Lie derivative (2.52), we find that X ABC should be of the form X ABC = Θ A α ( t α ) BC − (cid:2) β ( t α ) AD ( t α ) BC + δ DA δ CB (cid:3) ϑ D , (3.3)where the embedding tensors are given by the Weizenb¨ock connection,Θ A α ≡ P A α B β Ω B β , ϑ A = (1 + β ) Ω A − β Ω D α ( t α ) AD . (3.4)In general, the Weizenb¨ock connection Ω ABC on the right-hand side is not constant, but itturns out that the non-constant parts are canceled out on the right-hand side (see section 3.4).Then, in order to evaluate the structure constants X ABC , it is enough to know the value ofΩ
ABC at a certain point x . Namely, in the embedding tensors (3.4), we can replace Ω ABC with the constants F ABC ≡ Ω ABC (cid:12)(cid:12) x = x ≡ (cid:0) F A α t α + F A t (cid:1) BC . (3.5)In the following, we define the ExDA by requiring additional conditions on F ABC . The important requirement is that the ExDA contains an n -dimensional subalgebra g T a ◦ T b = f abc T c , (3.6)that satisfies the maximal isotropicity h T a , T b i = 0 (cid:2) h T A , T B i ≡ Y CDAB T C ⊗ T D (cid:3) . (3.7)Once the subalgebra g is fixed, as explained in section 2.1, we can decompose the generatorsof the duality group G such that the level-0 generators contain the GL( n ) generators K ab .15e can then decompose the R representation into representations of the subalgebra h (recallsection 2.2): { T A } = { T a , T ˜ C } where T ˜ C are called the dual generators. In general, we have T a ◦ T b = X abc T c + X ab ˜ C T ˜ C , (3.8)and the requirement (3.6) is equivalent to X abc = f abc , X ab ˜ C = 0 . (3.9)In order to consider the second condition, let us use an expression X ab ˜ C = (cid:2) δ Da δ βα + ( t β t α ) aD + χ β a χ D α (cid:3) F D β ( t α ) b ˜ C = (cid:2) δ D [ a | ( t α ) | b ] ˜ C + Y ˜ CDEb ( t α ) aE + χ α a χ D β ( t β ) b ˜ C (cid:3) F D α . (3.10)In general, the constants inside the square brackets do not vanish and in order to realize X ab ˜ C = 0 , we need to impose a condition on F A α . By considering ( R A ) aB = 0 , Y CDeb = 0(i.e., the maximal isotropicity), and χ K bc a = χ R I a = χ R A a = 0 , the (sufficient) condition is F A α t α = F Abc K bc + F A I R I + F A B R B (i.e., F A B = 0) . (3.11)Under this condition, we find X abc = 2 F [ ab ] c , (3.12)and the subalgebra g is a Lie algebra T a ◦ T b = − T b ◦ T a = f abc T c (cid:0) f abc = 2 F [ ab ] c (cid:1) . (3.13)Now let us explain the consequences of the condition (3.11) on the generalized framefields E AI . Initially, the generalized frame fields E AI were supposed to be a general elementof G × R + and then the Weizenb¨ock connection Ω ABC can be expanded as in Eq. (2.49).However, F ABC and Ω
ABC are related as F A α t α + F A t = (cid:2) Ω A α t α + Ω A t (cid:3) x = x = E AJ E BI ∂ J E I C (cid:12)(cid:12) x = x , (3.14)and Eq. (3.11) means that E AI should be generated by non-positive generators and R + , E AI = Π AB (e ( ˜ K + t )∆ ) BC E C I , E ≡ e − h I R I e − ( h R ) ab ˜ K ab , Π ≡ Y A e − π A R A , (3.15)where ˜ K ≡ ˜ K aa and the function ∆ is associated with the scale symmetry R + and thefunctions h and π parameterize a twist by the non-positive generators. Here, the generator˜ K ab ≡ K ab + β δ ab t , (3.16) Naively, it appears to be more natural to introduce ∆ as E AI = Π AB (e t ∆ ) BC E CI = e − ∆ Π AB E BI buthere we introduced ˜ K for later convenience (by following [4]). gl ( n ) algebra as K ab and the matrix representation has the form˜ K cd ≡ δ ca δ bd ∗ . (3.17)The details of ∆, π A , and E AI are explained later in section 3.3. Here, it is enough to knowthat ∆ and π A vanish at a certain point x :∆ | x = x = 0 , π A | x = x = 0 . (3.18)According to this property, we can easily compute the constants F ABC by using Eq. (3.14). E AI Another condition on F ABC comes from the section condition of ExFT. The derivative ∂ J onthe right-hand side of Eq. (3.14) should satisfy the section condition (2.2). As we explain insection 3.3, the generalized frame fields E AI are constructed from a group element g ≡ e x a T a where T a denotes the generators of the subalgebra g and x a (= δ ai x i ) are the corresponding n coordinates (which is sometime called the physical coordinates). Accordingly, here we supposethat E AI depends only on the physical coordinates x i , and then the section condition (2.2) istrivially satisfied because Y ijKL = 0 . Then we find F ABC = E AJ E BI ∂ j E I C (cid:12)(cid:12) x = x = E Aj E BI ∂ j E I C (cid:12)(cid:12) x = x = δ dA F dB C , (3.19)where we have used that E Aj = δ dA E dj . By using Eq. (3.15), F aBC can be parameterized as F aBC = (cid:2) k ade ˜ K de + f a I R I + f a A R A − Z a ( ˜ K + t ) (cid:3) BC , (cid:0) k ade ˜ K de + f a I R I (cid:1) BC ≡ E BI D a E I C (cid:12)(cid:12) x = x , f a A ≡ D a π A | x = x , Z a ≡ D a ∆ | x = x , (3.20)where D A ≡ E AI ∂ I . This can be equivalently expressed as F a α t α = ( k abc − Z a δ cb ) K bc + f a I R I + f a A R A , F a = β ( k abb − Z a δ bb ) − Z a . (3.21)In the next subsection, we shall use a short-hand notation { R ˆ A } = { R I , R A } . (3.22) Now, using the above setup, we can write down the structure constants X ABC of the ExDA.By using the formula (3.4) and Eq. (3.21), the embedding tensors are obtained asΘ A α t α = ( f Acd − Z [ A δ dc ] ) K cd + ( k bcd − Z b δ dc ) (cid:2) ( ˜ K cd R A ) Ab + χ K cd A χ b A (cid:3) R A + f A ˆ A R ˆ A − f c B ( R B ) Ad K cd + f b ˆ B ( R ˆ B R A ) Ab R A + f b ˆ B χ ˆ B A χ b A R A + β β (cid:2) ( f bcc − Z [ b δ cc ] ) ( t α ) Ab − f b B ( t α R B ) Ab (cid:3) t α , (3.23) ϑ A = β ( f Abb − Z [ A δ bb ] ) − β f b A ( R A ) Ab − (1 + β ) Z A , (3.24)17here f Abc ≡ δ dA f dbc , f abc ≡ k [ ab ] c , f A ˆ B ≡ { f A I , f A B } ≡ δ cA f c ˆ B , and Z A ≡ δ bA Z b . Substitutingthese embedding tensors into Eq. (3.3), we obtain the structure constants of the ExDA as X ABC = f A ˆ A ( R ˆ A ) BC + (cid:2) f Ade − f d A ( R A ) Ae − Z A δ ed (cid:3) ( K de ) BC + (cid:8) ( k def − Z d δ fe ) (cid:2) ( ˜ K ef R A ) Ad + χ K ef A χ d A (cid:3) + f d ˆ B (cid:2) ( R ˆ B R A ) Ad + χ ˆ B A χ d A (cid:3) + ( R A ) Ad Z d (cid:9) ( R A ) BC + (cid:2) β ( f Add − Z [ A δ dd ] ) − β f d A ( R A ) Ad − (1 + β ) Z A (cid:3) ( t ) BC . (3.25)In the second line, the constants k abc appear without antisymmetrizing the lower indices.However, we find a general property( ˜ K ef R A ) Ad + χ K ef A χ d A = ( ˜ K [ ef R A ) Ad ] + χ K [ ef A χ d ] A , (3.26)and k def can be replaced by its antisymmetric part k [ de ] f = f def : X ABC = f A ˆ A ( R ˆ A ) BC + (cid:2) f Ade − f d A ( R A ) Ae (cid:3) ( ˜ K de ) BC + (cid:8) ( f def − Z [ d δ fe ] ) (cid:2) ( ˜ K ef R A ) Ad + χ K ef A χ d A (cid:3) + f d ˆ B (cid:2) ( R ˆ B R A ) Ad + χ ˆ B A χ d A (cid:3) + ( R A ) Ad Z d (cid:9) ( R A ) BC − Z A ( ˜ K + t ) BC . (3.27)This is the general formula for the structure constants.For example, in DFT where β = 0 and Z a = 0, Eq. (3.27) reduces to X ABC = f Ad d ( R d d ) BC + (cid:2) f Ade − f df f ( R f f ) Ae (cid:3) ( K de ) BC + f def ( K ef R g g ) Ad ( R g g ) BC . (3.28)Using the matrix representations of the o ( d, d ) generators, K cd = δ ca δ bd − δ ad δ cb , R c c = δ c c ab , R c c = δ abc c , (3.29)which satisfy [ K ab , R c c ] = 2 δ c c bd R ad , [ K ab , R c c ] = − δ adc c R bd , [ R a a , R b b ] = 4 δ [ b [ a K b ] a ] , (3.30)we obtain the standard Drinfel’d algebra as T A ◦ T B = X ABC T C , T a ◦ T b = f abc T c , T a ◦ T b = f abc T c − f acb T c = − T b ◦ T a , T a ◦ T b = f cab T c . (3.31)Similarly, by considering the U -duality group G = E n ( n ) , we can reproduce the EDA fromthe general formula Eq. (3.27). In sections 5 and 6, this is worked out from the M-theoryand the type IIB perspective, respectively. Before getting into the details about the EDA, wediscuss a few more general aspects of ExDA. In n = 8, χ α A is indispensable to maintain this property. .2 Leibniz identities In this section, we consider the Leibniz identity T A ◦ ( T B ◦ T C ) = ( T A ◦ T B ) ◦ T C + T B ◦ ( T A ◦ T C ) . (3.32)In terms of the structure constants X ABC , this is equivalently expressed as X AC D X BDE − X BC D X ADE + X ABD X DC E = 0 . (3.33)If we define X A ≡ b Θ A + ϑ A t , b Θ A ≡ b Θ A α t α ≡ (cid:2) Θ A α − β ( t α ) AB ϑ B (cid:3) t α , (3.34)the Leibniz identity can also be expressed as[ X A , X B ] = − X ABC X C (cid:0) ⇒ Z ABC X C = 0 (cid:1) , (3.35)where Z ABC ≡ X ( AB ) C . Since t commutes with all generators, the left-hand side is anelement of the duality algebra. Then we can decompose (3.35) into the following two identities:[ b Θ A , b Θ B ] = − X ABC b Θ C , (3.36) X ABC ϑ C = 0 . (3.37)As was studied in [3,4,9], the Leibniz identity consists of several types of conditions on thestructure constants f abc , f a I , f a B , and Z a . Here, we show that the identity Eq. (3.36) containsthe so-called the cocycle conditions and the fundamental identities. They are not all of theconstraints on the structure constants and, in sections 5 and 6, some of additional constraintsfor the EDA are reproduced from Z ABC X C = 0 in Eq. (3.35). Let us consider a particular component of the Leibniz identity,[ b Θ a , b Θ b ] = − f abc b Θ c . (3.38)For convenience, we decompose the embedding tensor into the level-0 part b Θ (0) a (that is alinear combination of K ab and R I ) and the negative-level part e Θ a : b Θ a = b Θ (0) a + e Θ a . (3.39)Then the level-0 part of Eq. (3.38) gives[ b Θ (0) a , b Θ (0) b ] = − f abc b Θ (0) c . (3.40)19n particular, if we consider the coefficients of K ab , we find f acd f bde − f bcd f ade = − f abd f dce , f abc Z c = 0 . (3.41)The former is the standard Bianchi identity f [ abd f c ] de = 0 and the latter was found in [4].When there are additional level-0 generators { R I } (e.g., the type IIB case), we find additionalidentities that come from Eq. (3.40).If we consider the negative-level part of Eq. (3.38), we obtain[ b Θ (0) a , e Θ b ] − [ b Θ (0) b , e Θ a ] + f abc e Θ c + [ e Θ a , e Θ b ] = 0 . (3.42)To clarify the meaning of Eq. (3.42), we introduce a notation that is similar to the one used inthe Lie algebra cohomology. An n -cochain of g is a skew-symmetric linear map f : g n → N − where N − denotes the subalgebra generated by the negative-level generators R A , f ( x , . . . , x n ) = x a · · · x a n n f a ··· a n A R A , (3.43)where x i = x ai T a and f a ··· a n A are certain constants satisfying f a ··· a n A = f [ a ··· a n ] A . We denotethe vector space of n -cochains by C n and define the coboundary operator δ n : C n → C n +1 as δ f ( x ) ≡ x a (e ad f − b Θ (0) a = x · f + (cid:2) f, (cid:2) f, x a b Θ (0) a (cid:3)(cid:3) + · · · (cid:0) f ≡ f A R A (cid:1) , (3.44) δ f ( x, y ) ≡ x · f ( y ) − y · f ( x ) − f ([ x, y ]) − [ f ( x ) , f ( y )] , (3.45) δ f ( x, y, z ) ≡ x · f ( y, z ) + y · f ( z, x ) + z · f ( x, y ) − f ([ x, y ] , z ) − f ([ y, z ] , x ) − f ([ z, x ] , y ) − [ f ( x ) , f ( y, z )] − [ f ( y ) , f ( z, x )] − [ f ( z ) , f ( x, y )] , (3.46)where x · f ( y , · · · , y n ) ≡ (cid:2) f ( y , · · · , y n ) , x a b Θ (0) a (cid:3) . (3.47)We can check δ n +1 δ n = 0 by using Eq. (3.40), namely, x · (cid:0) y · f ( z , . . . , z n ) (cid:1) − y · (cid:0) x · f ( z , . . . , z n ) (cid:1) = [ x, y ] · f ( z , . . . , z n ) . (3.48)The most non-trivial coboundary operator is δ , which is defined so as to satisfy δ δ = 0 .By using the above definitions, the Leibniz identity (3.42) is interpreted as the cocyclecondition [4, 9], δ f ( x, y ) = x · f ( y ) − y · f ( x ) − f ([ x, y ]) − [ f ( x ) , f ( y )] = 0 , (3.49)where we have identified the constants f a A as the dual structure constants, and thus f ( x ) = x a f a A R A = x a e Θ a . (3.50)In summary, the Leibniz identity (3.38) can be rewritten as Eqs. (3.48) and (3.49).20n DFT, N − is Abelian and we have [ f ( · · · ) , f ( · · · )] = 0 and (cid:2) f, (cid:2) f, x a b Θ (0) a (cid:3)(cid:3) = 0 . Thenthe coboundary operators reduce to the usual ones. In particular, we have x · f ( y ) = − x a y b f ac [ d f bd ] c R d d , (3.51)and the cocycle condition (3.49) becomes2 f ac [ d f bd ] c − f bc [ d f ad ] c + f abc f cd d = 0 . (3.52)For completeness, it may be interesting to find the definition of the coboundary operators δ n for n ≥ δ n +1 δ n = 0. We find that the coboundary operator can be generallydefined as δ n f ( x , . . . , x n +1 ) ≡ X i ( − i − x i · f ( x , . . . , ˆ x i , . . . , x n +1 )+ X i The first property follows from the definition (3.65). By choosing g = e , where e is the unitelement of the physical subgroup G , we find M AB ( e ) = δ BA ⇔ π A | g = e = 0 , ∆ | g = e = 0 , A AB | g = e = δ BA . (3.69)They reproduce Eq. (3.18) if we denote the coordinate value of the unit element e as x . Algebraic identity The second property comes from the Leibniz identity, which shows g ⊲ ( T A ◦ T B ) = ( g ⊲ T A ) ◦ ( g ⊲ T B ) . (3.70)In terms of M AB , this reads( M − ) DC X ABD = ( M − ) AD ( M − ) BE X DEC . (3.71)From this identity, we find non-trivial algebraic identities for A AB , π A , and ∆ .For example, in DFT, the algebraic identity (3.71) gives f abc = a aa ′ a bb ′ ( a − ) c ′ c f a ′ b ′ c ′ , f d [ ab π c ] d + f de [ a π b | d | π c ] e = 0 ,f abc = a ad ( a − ) eb ( a − ) f c f def + 2 f ad [ b π c ] d , (3.72)which are presented in Appendix A of [59]. Differential identity The third property follows from the identity( hg ) − ⊲ T A = g − ⊲ ( h − ⊲ T A ) , (3.73)which can be shown as follows. Since g is a Lie algebra, for g = e g a T a and h = e h a T a , we have hg = e h a T a e g b T b = e ( hg ) a T a , (3.74)where ( hg ) a can be determined by using the Baker–Campbell–Hausdorff formula. Since X a satisfies the same algebra as − T a , we find (cid:0) e − ( hg ) a X a (cid:1) AB T B = (cid:0) e − h a X a e − g b X b (cid:1) AB T B , (3.75)and this shows the identity (3.73). 23n terms of M AB , the identity (3.73) is equivalent to M AB ( hg ) = M AC ( h ) M C B ( g ) . (3.76)For example, in DFT, this relation gives a ab ( hg ) = a ac ( h ) a cb ( g ) , π ab ( hg ) = π ab ( h ) + π cd ( g ) ( a − ) ca ( h ) ( a − ) db ( h ) . (3.77)The second equation shows that the bi-vector π ij ≡ π ab e ia e jb is multiplicative (see e.g., [60]).We thus consider that the relation (3.76) defines that the multi-vectors π A are multiplicative.Now, we can derive a differential identity from the identity (3.76). Supposing h = e + ǫ a T a ,where e is the unit element of G and ǫ a are infinitesimal parameters, we have M AB ( h ) = δ BA − ǫ c X cAB . (3.78)Then, since an infinitesimal left multiplication g ( x ) → (1 + ǫ a T a ) g ( x ) corresponds to theinfinitesimal coordinate transformation δx i = ǫ a e ia , we obtain the differential identity D c M AB ( g ) = lim ǫ a → M AB ( hg ) − M AB ( g ) ǫ a = − X cAD M DB ( g ) , (3.79) ⇔ M AD D c ( M − ) DB = X cAB . (3.80)Recalling the decomposition M = Π e ( ˜ K + t )∆ A , we find that the block-diagonal componentsof this relation (i.e., matrix elements ( · · · ) AB with A and B having the same level) give A D c A − = f cde ˜ K de + f c I R I , D a ∆ = Z a . (3.81)The other components give the differential identity for Π as (cid:0) Π D c Π − (cid:1) AB = ( b Θ c ) AB − (cid:0) Π b Θ (0) c Π − (cid:1) AB . (3.82)They determine the derivatives of A AB , ∆, and π A . Another useful expression that followsfrom (3.82) and the algebraic identity is v ka E I A (e − K ∆ ∂ k Π − Π e K ∆ ) AB E BJ = ˆ V I A (cid:0) f a A R A (cid:1) AB ˆ V BJ . (3.83)In DFT, the differential identity (3.82) gives D c π ab = f cab − f cd [ a π b ] d . (3.84)This leads to the condition that π is multiplicative, £ v a π mn = f abc v mb v nc . (3.85)Here, we used the identities (3.72), but Eq. (3.85) can be directly obtained from Eq. (3.83).In the EDA, we use Eq. (3.83) when we derive the relations similar to Eq. (3.85).24 .4 Generalized parallelizability Now we are ready to show that the generalized fluxes E AI defined in Eq. (3.68) indeed satisfiesthe algebra ˆ £ E A E BI = − X ABC E BI . (3.86)This property for the Drinfel’d algebra was shown in [27] by using DFT, and the same propertyfor the E n ( n ) EDA ( n ≤ 6) was shown in [3, 4, 9]. There, the proof depends on the detailsof the duality group G and the choice of the subalgebra g . Consequently, the proof becomesmore complicated as the dimension of the ExDA becomes larger. However, the general proofto be presented here is simple and does not depend on G and the choice of g . It can be appliedeven to the E EDA, both in the M-theory and the type IIB picture.Before going into the details of the technical computation, let us explain the outline of ourproof. The point is that, as we show below, the Weizenb¨ock connectionΩ ABC = E AI E BK ∂ I E K C , (3.87)can be expressed as a duality twist of some functions F ABC :Ω ABC = M AD M BE ( M − ) F C F DEF , (3.88)where M AB is the matrix defined in Eq. (3.65). Since M AB ∈ G × R + and the generalizedfluxes X ABC are constructed from Ω ABC and G -invariant tensors (recall Eqs. (3.3) and (3.4)),this means that the generalized fluxes have the form X ABC = M AD M BE ( M − ) F C X DEF , (3.89)where X ABC are the generalized fluxes (2.52) with Ω ABC replaced with F ABC .As we show below, the non-constant part of F ABC does not contribute to X ABC and thegeneralized fluxes X ABC are constant. Evaluating the constants X ABC at a particular point x = x (where M AB = δ BA and X ABC = X ABC ), we can easily see X ABC = X ABC . Then, thealgebraic identity (3.71) shows X ABC = M AD M BE ( M − ) F C X DEF = X ABC . (3.90)By recalling the definition of the generalized fluxes, this is equivalent to the desired relationˆ £ E A E BI = − X ABC E BI = − X ABC E BI . (3.91)Thus, our remaining tasks are to derive Eq. (3.88) with some functions F ABC and to showthat X ABC are constant. 25sing the definition of the generalized frame fields given in Eq. (3.68), and the definitionof the Weizenb¨ock connection (3.87), we obtainΩ ABC = M Ad (cid:8) M BE ( M − ) F C ˆΩ dEF + ( a − ) de (cid:0) M D e M − (cid:1) BC (cid:9) , (3.92)ˆΩ ABC ≡ ˆ V AI ˆ V BK ∂ I ˆ V K C . (3.93)Here, recalling Eq. (3.61) and using ˆ V Ai = δ bA v ib , we haveˆΩ ABC = δ eA k ef g ˜ K f g , k abc ≡ v kb v ia ∂ i ℓ ck . (3.94)In general, k abc is non-constant, but its antisymmetric part is given by the structure constants k [ ab ] c = − f abc . Then, using the differential identity (3.80), we can rewrite Eq. (3.92) asΩ ABC = M Ad (cid:2) k def ( M ˜ K ef M − ) BC + ( a − ) de X eBC (cid:3) . (3.95)Then, by using k [ ab ] c = − f abc , M Ab = Π Ac a cb , and the identity( a − ) ad X dB C = M BD ( M − ) EC X aDE , (3.96)which can be obtained from the algebraic identity (3.71), we obtainΩ ABC = M Ad M BB ′ ( M − ) C ′ C F aBC , (3.97) F aBC ≡ (cid:0) k ( ad ) e − f ade (cid:1) ( ˜ K de ) BC + X aBC . (3.98)Finally, by introducing F ABC ≡ δ dA F dBC , (3.99)the Weizenb¨ock connection can be expressed in the desired formΩ ABC = M AD M BE ( M − ) F C F DEF . (3.100)Now, let us explain the constancy of X ABC . The generalized fluxes X ABC are constructedfrom F ABC but its non-constant part is coming only from the symmetric part k ( ab ) c of k abc .Namely, the non-constant part of X ABC becomes X ABC | non-constant = k ( de ) f (cid:2) ( ˜ K ef R A ) Ad + χ K ef A χ d A (cid:3) ( R A ) BC . (3.101)Using the identity (3.55), we find that this vanishes. Namely, we have shown that X ABC isconstant, and thus X ABC = X ABC .This completes the proof that for any choice of the maximally isotropic subalgebra g , theExDA always gives a generalized parallelizable space.26 .4.1 Nambu–Lie structures In ExFT, the generalized fluxes are decomposed into representations of the subgroup h . Themost familiar example is the case of DFT, where X ABC are decomposed as H abc ≡ X abc , F abc ≡ X abc , Q abc ≡ X abc , R abc ≡ X abc , (3.102)because X ABC are totally antisymmetric. Using the generalized frame fields E a = e a , E a = − π ab e b + r a (cid:0) r a ≡ r am d x m (cid:1) , (3.103)we can compute the fluxes as H abc = 0 , F abc = f abc , Q abc = D a π bc + 2 f ad [ b π c ] d ,R abc = 3 π d [ a D d π bc ] + 3 f d d [ a π b | d | π c ] d . (3.104)Then, the generalized parallelizability means that Q abc = D a π bc + 2 f ad [ b π c ] d = f abc , R abc = 3 π d [ a D d π bc ] + 3 f d d [ a π b | d | π c ] d = 0 . (3.105)Namely, the dual structure constants are identified as the Q -flux (also known as the globallynon-geometric flux) and the absence of the R -flux (called the locally non-geometric flux) givesa differential identity for π ab . For the bi-vector field π mn = π ab e ma e nb , the latter reads π qm ∂ q π np + π qn ∂ q π pm + π qp ∂ q π mn = 0 , (3.106)which shows that π mn is a Poisson tensor.In a general ExFT, the correspondent of the R -fluxes are X ˚ A ˚ Bc and we observe that theyalways vanish, X ˚ A ˚ Bc = X ˚ A ˚ Bc = 0 . (3.107)The computation of the flux X ˚ A ˚ Bc is very hard in the E EFT, and we do not compute theexplicit components in this paper. Here we just show an example, a locally non-geometric flux X a a b b c in M-theory picture. Under some algebraic identities, this is expressed as X a a b b c = − π a a d ∇ d π b b c + 3 π d [ b b ∇ d π c ] a a + f d d [ a π a ] b b cd d = 0 , (3.108)where ∇ b π a a a ≡ D b π a a a − f bc [ a | π c | a a ] . This may be further simplified by using certainrelations among π and the structure constants, but it is obvious that this is a natural extensionof the R -flux. As noted in [3], at least if we assume f abc = 0, this is precisely the conditionthat the π i i i is the Nambu–Poisson tensor. Suggested by this, we regard Eq. (3.107) asa definition that the multi-vectors π are the generalized Nambu–Poisson tensors. Moreover,combining this with the fact that π are multiplicative, we call π the Nambu–Lie structure [61].27 .5 Generalized classical Yang–Baxter equation As discussed in section 3.2.1, the Leibniz identity requires that the dual structure constants f a A satisfy the cocycle condition (3.49), δ f ( x, y ) = 0. This condition is trivially satisfied byassuming that the cocycle f ( x ) is a coboundary [4, 9], f ( x ) = δ r ( x ) ≡ x a (e ad r − b Θ (0) a (cid:0) r ≡ ˜ r A R A (cid:1) . (3.109)By recalling f ( x ) = x a e Θ a and b Θ a = b Θ (0) a + e Θ a , this coboundary ansatz corresponds to b Θ a = e ad r b Θ (0) a . (3.110)The right-hand side can be rewritten as b Θ a = e ad r b Θ (0) a = R b Θ (0) a R − , R ≡ Y A e r A R A , (3.111)where r A is a redefinition of ˜ r A . By comparing this with the general expression b Θ a = b Θ (0) a + f a A R A , the dual structure constants f a A can be expressed symbolically as f a A R A = R b Θ (0) a R − − b Θ (0) a ≡ f a A R A . (3.112)For example, in DFT, we have b Θ (0) a = f abc K bc and r = r a a R a a , and the coboundaryansatz becomes b Θ a = f abc K bc − f ac [ b r b ] c R b b . (3.113)Comparing this with the Drinfel’d algebra b Θ a = f abc K bc + f ab b R b b , (3.114)we can identify the dual structure constants as f ab b = 2 f ac [ b | r c | b ] ≡ f ab b . (3.115)Since all of the dual structure f a B are replaced by f a B , we obtain an ExDA whose structureconstants X ABC are characterized by f abc , f a I , Z a , and r A . We call this the coboundary ExDA.The cocycle conditions are trivially satisfied but other Leibniz identities are not satisfied ingeneral, and we need to impose further conditions on r A . Below, following the approachdiscussed in [3], we explain the procedure to find such additional conditions on r A .Let us note that Eq. (3.111) can be expressed as X aBC = R aD R BE ( R − ) F C X (0) DEF , X (0) ABC ≡ ( X r =0 ) ABC , (3.116)where R AB is the matrix representation of R in the R representation. This suggests us toextend this relation to define a new Leibniz algebra twisted by R , X ABC ≡ R AD R BE ( R − ) F C X (0) DEF . (3.117)28he untwisted structure constants X (0) ABC satisfy the Leibniz identity (when the Bianchi iden-tities (3.40) are satisfied) and it is clear that X ABC also satisfy the Leibniz identity (3.33).However, the algebra defined by the structure constants X ABC is not an ExDA in general.Only the particular components, namely, X aBC coincide with X aBC of an ExDA. Thus, toensure that X ABC are the structure constants of a certain ExDA, we require X ˜ ABC = X ˜ ABC . (3.118)This requires certain conditions on r A and if these are satisfied, we obtain a coboundary ExDAthat automatically satisfies the Leibniz identity.For example, in the case of DFT, using R = e r a a R a a we find X aBC = 2 f df [ a r e ] f ( K de ) BC + f d d a ( R d d ) BC . (3.119) X aBC = r af f fde ( R K de R − ) BC + f d d a ( R R d d R − ) BC = 2 f df [ a r e ] f ( K de ) BC + f d d a ( R d d ) BC + r d [ a r | d | e f d d f ] ( R ef ) BC . (3.120)Then the requirement (3.118) is equivalent to the classical Yang–Baxter equation (CYBE), r d a r d e f d d f + r d e r d f f d d a + r d f r d a f d d e = 0 . (3.121)The definition of the generalized CYBE (3.118) is very simple, but in practice, we findthat it is difficult to identify the full set of the independent conditions on r A . In general, wecan express X r and X R as X ˜ A = · · · |{z} non-negative-level generators − β f d A ( R A ) ˜ Ad t , (3.122) X ˜ A = · · · |{z} non-negative-level generators − β f d A ( R A ) ˜ Ad t + CYBE ˜ A A R A + CYBE ˜ A t , (3.123)whereCYBE ˜ A B ≡ R ˜ Ab f b B + R ˜ AB (cid:8)(cid:0) f def − Z [ d δ fe ] (cid:1) (cid:2) ( ˜ K ef R A ) Bd + χ K ef B χ d A (cid:3) + f d I ( R I R A ) Bd + χ I B χ d A (cid:1) + ( R A ) Bd Z d (cid:9) [ R R A R − ] B , (3.124)CYBE ˜ A ≡ β f c B ( R B ) ˜ Ac + R ˜ Ab (cid:2) β ( f bcc − Z b δ cc ) − Z b (cid:3) , (3.125)and [ · · · ] B denotes the coefficient of R B : namely, R R A R − = · · · + [ R R A R − ] B R B . Then thecondition (3.118) at least requiresCYBE ˜ A B = 0 , CYBE ˜ A = 0 . (3.126)In general, terms including the non-negative-level generators in X ABC and X ABC do notcoincide and we find additional conditions, but extensions of the quadratic equation (3.121)are contained in CYBE ˜ A B = 0 as we see in the case of the EDA.29 Non-Abelian duality In this section, we explain the procedure of the non-Abelian duality. For a given ExDA, wechoose a maximally isotropic subalgebra g and construct the generalized frame fields E AI asexplained in the previous section. For simplicity, if we suppose θ A = 0, the generalized framefield has unit determinant and then we define the generalized metric as M IJ = E I A E J B ˆ M AB (det ˆ M = 1) . (4.1)In general, ˆ M AB can depend on the external coordinates (see [62] for details), but here wesuppose that it is constant for simplicity and consider only the internal part of the spacetime.According to the flux formulation [63], the equations of motion of the ExFT are covariantequations for the fluxes X ABC and ˆ M AB . In our situation, the generalized fluxes areconstants X ABC and the equations of motion are algebraic equations for X ABC and ˆ M AB . Ifwe find a solution for ˆ M AB , the background (4.1) is a solution of ExFT. Since M IJ dependsonly on the physical coordinates x i , it is also a solution of the standard supergravity. In general, the ExDA has another maximally isotropic subalgebra g ′ whose generators aredenoted as T ′ a . We also identify the full set of generators { T ′ A } = { T a , · · · } such that T ′ A satisfy an ExDA. This is just a redefinition of generators T ′ A = C AB T B ( C AB : constant) , (4.2)and the structure constants of the new algebra are X ′ ABC = C AA ′ C BB ′ ( C − ) C ′ C X A ′ B ′ C ′ . (4.3)Using the new set of generators T ′ A , we can construct new generalized frame fields E ′ AI whichdepend on the physical coordinates in the primed system x ′ i . Here, the dimension of thephysical space can be different, in which case C AB is not an element of the duality group G . In general, the frame fields E ′ AI are completely different from the original ones E AI and theyare not related by C AB : E ′ AI = C AB E BI . However, the generalized fluxes in the original andthe dualized frame are related as in Eq. (4.3) and since the equations of motion of ExFT arecovariant equations of the fluxes and the constant matrix ˆ M AB , by introducingˆ M ′ AB ≡ C AA ′ C BB ′ ˆ M A ′ B ′ , (4.4)the new background M ′ IJ = E ′ I A E ′ J B ˆ M ′ AB , (4.5)is a solution of ExFT. This is a basic procedure to perform the non-Abelian duality. In DFT, we also have the dilaton and we need to consider the corresponding flux F A as well (see [62]). When θ A does not vanish, it will be a solution of a certain gauged supergravity. The linear map of [64], which relates the M-theory and the type IIB picture, is such an example. .1 Yang–Baxter deformation Let us consider the Yang–Baxter deformation [65] as a particular class of non-Abelian duality.In this case, we assume that the original ExDA is a semi-Abelian algebra, meaning thatthe dual structure constants f a A are absent. Then the generalized frame fields are simplygiven by E AI = e ( ˜ K + t )∆ E AI . We now consider the redefinition (4.2) with C AB = R AB where R is the twist matrix defined in Eq. (3.111). As long as r A satisfies the generalizedCYBE, the redefined generators T ′ A satisfy a coboundary ExDA. Using the ExDA, we canstraightforwardly compute the generalized frame fields E ′ AI . Since f abc , f a I , and Z a are notdeformed under the redefinition T ′ A = R AB T B , we find E ′ AI = Π ′ AB E BI , (4.6)where E AI are the original generalized frame fields. Then we obtain a new solution of ExFT, M ′ IJ ≡ E ′ I A E ′ J B ˆ M ′ AB = U I K U J L M KL , (4.7) U I J ≡ E I A ( Π ′− R ) AB E BJ . (4.8)For example, in DFT, we obtain( U I J ) ≡ δ nm π ′ ab + r ab ) e ma e nb δ mn . (4.9)For a general ExDA, it is a hard task to obtain π ′ A because we need to compute theexponential action (3.65). However, the usefulness of the Yang–Baxter deformation is that wecan generally express π A in terms of the classical r -matrix. For example, in DFT, we find π ′ mn = r ab ( v ma v nb − e ma e nb ) . (4.10)Indeed, this vanishes at the unit element x = x and also satisfies the differential equation(3.85). Then, the matrix U I J is simplified as( U I J ) ≡ δ nm ρ mn δ mn , ρ mn ≡ r ab v ma v nb . (4.11)Thus, for a given solution r ab of CYBE (3.121), we can easily obtain the deformed background M ′ IJ simply by acting the coordinate-dependent matrix U I J .The same story can be extended to a general coboundary ExDA. In sections 5 and 6, wefind the explicit form of the Nambu–Lie structures π ′ for coboundary EDAs. We then findthat the matrix U I J has a very simple form. Again, U I J is parameterized by the multi-vectors ρ which is given by the classical r -matrix r A and the left-invariant vector fields v ia . Here, we suppose that the structure constants f a I and Z a can be present. Exceptional Drinfel’d algebra: M-theory section As explained in section 2.2, the R representation in the M-theory picture is decomposed as( T A ) = (cid:0) T a , T a a √ , T a ··· a √ , T a ··· a ,a ′ √ , T ,a a a √ , T ,a ··· a √ , T , ,a (cid:1) , (5.1)where 8 denotes eight totally antisymmetric indices 1 · · · 8, e.g., T ,a a a = T ··· ,a a a . Whenwe consider the case n = 7, generators including eight indices automatically disappear, andwhen we consider n = 6, T a ··· a ,a ′ additionally disappears because it requires seven antisym-metric indices. Then we find that the dimension of the EDA coincides with the dimension ofthe R representation given in Table A.1.Using the general results given in the previous section, the embedding tensors for E n ( n ) EDA ( n ≤ 8) in the M-theory picture are obtained as follows: X a = f ab b b R b b b + f ab ··· b R b ··· b + f ab ··· b ,c R b ··· b ,c + f abc ˜ K bc − Z a ( ˜ K + t ) , (5.2) X a a = − f cda a ˜ K cd − f c c [ a R a ] c c + 3 Z d R da a , (5.3) X a ··· a = − f cda ··· a ˜ K cd − f b [ a a a R a a ] b − f c c [ a R a ··· a ] c c + 6 Z b R ba ··· a , (5.4) X a ··· a ,a ′ = − (cid:0) f cda ··· a ,a ′ − f ca ··· a a ′ ,d (cid:1) ˜ K cd + 7 f b [ a ··· a R a ] a ′ b − f b [ a ··· a R a a ′ ] b + 21 f ba ′ [ a a R a ··· a ] b + 14 f b [ a a a R a ··· a a ′ ] b − f b b [ a R a ··· a ] b b ,a ′ + f bca ′ R a ··· a b,c + Z b (cid:0) R ba ··· a ,a ′ + R a ··· a a ′ ,b (cid:1) , (5.5) X a ··· a ,a ′ a ′ a ′ = 3 f ba ··· a , [ a ′ R a ′ a ′ ] b − f ba ′ a ′ a ′ [ a a a R a ··· a ] b + 3 f b [ a ′ a ′ | b R a ··· a , | a ′ ] + f ba ′ a ′ a ′ R a ··· a ,b , (5.6) X ,a ··· a = − f b , [ a R a ··· a ] b + 6 f b [ a ··· a | b R , | a ] + f ba ··· a R ,b , (5.7) X , ,a = 2 f d ,a R ,d + f d ,d R ,a . (5.8)In particular we have ϑ a = β f abb − (1 + β n ) Z a , ϑ a a = − β f bba a , ϑ a ··· a = − β f bba ··· a ,ϑ a ··· a ,a ′ = − β (cid:0) f bba ··· a ,a ′ − f ba ··· a a ′ ,b (cid:1) , ϑ , = ϑ , = ϑ , , = 0 . (5.9)By using these, we can easily write down the explicit form of the EDA, T A ◦ T B = ( X A ) BC T C . (5.10)We note that the vector Z a corresponds to − L a / − Z a / n ≤ T a ◦ T b = f abc T c ,T a ◦ T b b = f ab b c T c + 2 f ac [ b T b ] c + 3 Z a T b b ,T a ◦ T b ··· b = − f ab ··· b c T c − f a [ b b b T b b ] − f ac [ b T b ··· b ] c + 6 Z a T b ··· b ,T a ◦ T b ··· b ,b ′ = 7 f a [ b ··· b T b ] b ′ − f ab ′ [ b b T b ··· b ] − f ac [ b T b ··· b ] c,b ′ − f acb ′ T b ··· b ,c + 9 Z a T b ··· b ,b ′ ,T a a ◦ T b = − f ba a c T c + 3 f [ c c [ a δ a ] b ] T c c − Z c δ [ cb T a a ] ,T a a ◦ T b b = − f ca a [ b T b ] c − f c c [ a T a ] b b c c + 3 Z c T a a b b c ,T a a ◦ T b ··· b = 5 f ca a [ b T b ··· b ] c − δ a a de f c c d T b ··· b [ c c ,e ] + 9 Z c T b ··· b [ a a ,c ] ,T a a ◦ T b ··· b ,b ′ = 7 f ca a [ b T b ··· b ] c,b ′ + f ca a b ′ T b ··· b ,c ,T a ··· a ◦ T b = f ba ··· a c T c + 10 f b [ a a a T a a ] + 20 f c [ a a a δ a b T a ] c (5.11)+ 5 f bc [ a T a ··· a ] c + 10 f c c [ a δ a b T a a a ] c c − Z c δ [ cb T a ··· a ] ,T a ··· a ◦ T b b = 2 f ca ··· a [ b T b ] c − f c [ a a a T a a ] b b c + 5 f c c [ a T a ··· a ] c c [ b ,b ] − Z c T ca ··· a [ b ,b ] ,T a ··· a ◦ T b ··· b = − f ca ··· a [ b T b ··· b ] c − f c [ a a a δ a a ] cd d e T b ··· b d d ,e ,T a ··· a ◦ T b ··· b ,b ′ = − f ca ··· a [ b T b ··· b ] c,b ′ − f ca ··· a b ′ T b ··· b ,c ,T a ··· a ,a ′ ◦ T b = − f c [ a ··· a δ a ] a ′ cbd d T d d − f ca ′ [ a a δ a ··· a ] cbd ··· d T d ··· d ,T a ··· a ,a ′ ◦ T b b = 7 f c [ a ··· a T a ] a ′ cb b − f ca ′ [ a a T a ··· a ] c [ b ,b ] ,T a ··· a ,a ′ ◦ T b ··· b = 21 f c [ a ··· a δ a ] a ′ cd d e T b ··· b d d ,e ,T a ··· a ,a ′ ◦ T b ··· b ,b ′ = 0 . We can easily reproduce the E EDA of [9] (up to sign convention) by a truncation.Now, let us consider the Leibniz identities. If we define f ( x ) ≡ x a f ab b b R b b b , f ( x ) ≡ x a f ab ··· b R b ··· b ,f , ( x ) ≡ x a f ab ··· b ,b ′ R b ··· b ,b ′ , (5.12)the cocycle conditions (3.49) are decomposed as follows:d f ( x, y ) = 0 , (5.13)d f ( x, y ) − [ f ( x ) , f ( y )] = 0 , (5.14)d f , ( x, y ) − [ f ( x ) , f ( y )] − [ f ( x ) , f ( y )] = 0 , (5.15)where d f ∗ ( x, y ) ≡ x · f ∗ ( y ) − y · f ∗ ( x ) − f ∗ ([ x, y ]) ,x · f ∗ ( y ) ≡ x a [ f ∗ ( y ) , f abc K bc − Z a K ] . (5.16)33f we introduce r ≡ r a a a R a a a , r ≡ r a ··· a R a ··· a , r , ≡ r a ··· a ,a ′ R a ··· a ,a ′ , (5.17)and define R ≡ e r e r e r , , the coboundary ansatz (3.111) are decomposed as f ( x ) = d r ( x ) , (5.18) f ( x ) = d r ( x ) + (cid:2) r , f ( x ) (cid:3) , (5.19) f , ( x ) = d r , ( x ) + (cid:2) r , f ( x ) (cid:3) − (cid:2) r , (cid:2) r , f ( x ) (cid:3)(cid:3) , (5.20)where d r ∗ ( x ) ≡ x a (cid:2) r ∗ , f abc K bc − Z a K (cid:3) . (5.21)More explicitly, the cocycle conditions are expressed as0 = 6 f [ a | d [ c | f | b ] d | c c ] − f abd f dc c c − Z [ a f b ] c c c , (5.22)0 = 12 f [ a | d [ c | f | b ] d | c ··· c ] − f abd f dc ··· c − f [ a [ c c c f b ] c c c ] − Z [ a f b ] c ··· c , (5.23)0 = 16 f [ a | d [ c | f | b ] d | c ··· c ] ,c ′ + 2 f [ a | dc ′ f | b ] c ··· c ,d − f abd f dc ··· c ,c ′ − f [ a [ c c c f b ] c ··· c ] c ′ − Z [ a f b ] c ··· c ,c ′ , (5.24)and the coboundary ansatzes are f ab b b = 3 f ac [ b | r c | b b ] − Z a r b b b , (5.25) f ab ··· b = 6 f ac [ b | r c | b ··· b ] − f a [ b b b r b b b ] − Z a r b ··· b , (5.26) f ab ··· b ,b ′ = 8 f ac [ b | r c | b ··· b ] ,b ′ + f acb ′ r b ··· b ,c + 56 r [ b b b f ab ··· b ] b ′ − f a [ b b b r b b b r b b ] b ′ − Z a r b ··· b ,b ′ . (5.27)To compute the fundamental identities (3.56), we need the explicit components of thestructure constants X ˚ A ˚ B ˜ C . For simplicity, if we consider the case n ≤ 7, we find f cda a f db b b − f da a [ b f cb b ] d = f d d [ a f ca ] b b b d d + 3 f ca a b b b d Z d , (5.28) f cda ··· a f db ··· b − f da ··· a [ b f cb ··· b ] d = − f d [ a a a δ a a ] de e e f cb ··· b e e ,e , (5.29)which reduce to the known results [9] if we consider the reduction to n ≤ E n ( n ) EDA for n ≤ Z ABC X C = 0 of Eq. (3.35). For example, the non-vanishing components of Z a a bC are determined as Z a a bc c = 2 f c c [ a δ a ] b − δ a a b [ c Z c ] , (5.30)34nd then we find Z a a bC X C = Z a a bc c X c c = (cid:0) − f e e f + 6 Z e δ fe (cid:1) δ a a fb f cde e ˜ K cd = 0 , (5.31)where we have used the Bianchi identities f [ abd f c ] de = 0 and f abc Z c = 0. This is equivalent to f c c a f bdc c − Z c f badc = 0 . (5.32)As studied in [9], when n ≤ Z a = 0, this condition, the cocycle condition, the funda-mental identities, and the usual Bianchi identity f [ abd f c ] de = 0 are all of the Leibniz identities.In general, we will find additional identities, and in order to find the full set of identities,it may be useful to employ the results of the quadratic constraints studied in the gaugedsupergravity [48]. Here we do not study further on the Leibniz identities.For a given EDA, we can compute the adjoint action (3.65) to obtain the matrix M AB andusing that we can get the generalized frame fields E AI . We parameterize the matrix Π as Π = e − π a a a R a a a e − π a ··· a R a ··· a e − π a ··· a ,a ′ R a ··· a ,a ′ . (5.33)In the E case, the matrix size of E AI is large, so here we show the explicit form of thegeneralized frame fields for n ≤ E a = e a ,E a a = − π ba a e b + e − r a a ,E a ··· a = − (cid:0) π ba ··· a + 5 π [ a a a π a a ] b (cid:1) e b + 10 e − π [ a a a r a a ] + e − r a ··· a ,E a ··· a ,a ′ = − (cid:0) π b [ a ··· a π a a ] a ′ + 35 π a ′ [ a a π a a a π a a ] b (cid:1) e b − − π [ a ··· a r a ] a ′ + 105 e − π a ′ [ a a π a a a r a a ] + 21 e − r [ a ··· a π a a ] a ′ + e − r a ··· a ⊗ r a ′ , (5.34)where r a ··· a p ≡ r a ∧ · · · ∧ r a p ( r a ≡ r ai d x i ) and π A correspond to the Nambu–Lie structures.To find the explicit form of the differential identities (3.82), we use the identity,e − X D c e X = D c X − [ X, D c X ] + [ X, [ X, D c X ]] − [ X, [ X, [ X, D c X ]]] + · · · , (5.35)and then we obtain D a π = f a + f abc [ π , K bc ] − Z a π , D a ∆ = Z a ,D a π = f a + f abc [ π , K bc ] + [ π , f ] − Z a π ,D a π , = f a , + f abc [ π , , K bc ] + [ π , f ] + [ π , [ π , f ]] − Z a π , , (5.36)where f a ≡ f ( T a ), f a ≡ f ( T a ), f a , ≡ f , ( T a ), and π ≡ π b b b R b b b , π ≡ π b ··· b R b ··· b , π , ≡ π b ··· b ,b ′ R b ··· b ,b ′ . (5.37)35n components, they are expressed as D a π b b b = f ab b b + 3 f ac [ b | π c | b b ] − Z a π b b b , D a ∆ = Z a ,D a π b ··· b = f ab ··· b + 6 f ac [ b | π c | b ··· b ] − Z a π b ··· b − f a [ b b b π b b b ] ,D a π b ··· b ,b ′ = f ab ··· b ,b ′ + 8 f ac [ b | π c | b ··· b ,b ′ + f acb ′ π b ··· b ,c − Z a π b ··· b ,b ′ + 56 f ab ′ [ b ··· b π b b b ] + f a [ b b b π b b b π b b ] b ′ . (5.38)For the Nambu–Lie structures (e.g., π i i i = π a a a e i a e i a e i a ), we find £ v a π i i i = e − f ab b b v i b v i b v i b , £ v a ∆ = Z a , £ v a π i ··· i + 10 π [ i i i £ v a π i i i ] = e − f ab ··· b v i b · · · v i b , £ v a π i ··· i ,i ′ − (cid:0) π i ′ [ i ··· i + π i ′ [ i i π i i i (cid:1) £ v a π i i i ] = e − f ab ··· b ,b ′ v i b · · · v i b v i ′ b ′ , (5.39)by using Eq. (3.83). They are the properties for a general EDA.In particular, for a coboundary EDA, we can find a solution of the differential equations(5.39) that satisfies π = 0 at x = x , π i i i = r a a a (cid:0) e − v i a v i a v i a − e i a e i a e i a (cid:1) , (5.40) π i ··· i = r a ··· a (cid:0) e − v i a · · · v i a − e i a · · · e i a (cid:1) − π [ i i i r i i i ] , (5.41) π i ··· i ,i ′ = r a ··· a ,b ′ (cid:0) e − v i a · · · v i a v i ′ a ′ − e i a · · · e i a e i ′ a ′ (cid:1) + 56 e − r a a a π i ′ [ i ··· i v i a v i a v i ] a − e − r a a a π i ′ [ i i v i a v i a v i a r i i i ] + r i ′ [ i i r i i i π i i i ] , (5.42)where r i i i ≡ r a a a e i a e i a e i a and n ≤ U I J defined in (4.8).Since the above expression for π is intricate, one may consider that U I J is also complicated,but in fact U I J is surprisingly simple, U I J = (cid:0) e e − ρ e e − ρ e e − ρ , (cid:1) I J , (5.43)where we have defined ρ ≡ ρ i i i R i i i , ρ ≡ ρ i ··· i R i ··· i , and ρ , ≡ ρ i ··· i ,i ′ R i ··· i ,i ′ ,and the multi-vectors ρ are defined, for example, as ρ i i i ≡ r a a a v i a v i a v i a .This simple expression is not a coincidence and can be derived as follows. For convenience,let us define a matrix U AB ≡ E (0) A I U I J E (0) J B = ( Π − R ) AB , where E (0) A I ≡ E AI | π =0 . Thedifferential identity (3.80) shows( R U − e ( ˜ K + t )∆ A ) AD D c ( A − e − ( ˜ K + t )∆ U R − ) DB = X cAB , (5.44)and by using the coboundary ansatz, X cAB = ( R X (0) c R − ), this becomes( U − e ( ˜ K + t )∆ A ) AD D c ( A − e − ( ˜ K + t )∆ U ) DB = X (0) c . (5.45)36sing the differential identities Eq. (3.81), we can easily find a solution of this differentialequation. In fact, A − e − ( ˜ K + t )∆ U should coincide with A − e − ( ˜ K + t )∆ up to the left multi-plication by a constant matrix. The constant matrix can be found by considering U AB = R AB , A AB = δ BA , and ∆ = 0 at x = x , and then we get A − e − ( ˜ K + t )∆ U = R A − e − ( ˜ K + t )∆ . (5.46)This shows that U AB = (cid:0) A e ( ˜ K + t )∆ R e − ( ˜ K + t )∆ A − (cid:1) AB and then we find U I J = (Ω ∆ ) I J , (Ω ∆ ) I J ≡ ˆ V I A (cid:0) e ( ˜ K + t )∆ R e − ( ˜ K + t )∆ (cid:1) AB ˆ V BI . (5.47)This generally explains the simple result (5.43).Similar to the case of DFT, for a given solution of CYBE for r a a a , ρ a ··· a , and ρ a ··· a ,a ′ ,we can easily generate a new solution of EFT simply by multiplying the coordinate-dependenttwist matrix U I J to the original generalized metric.In the following, we consider the generalized CYBE by assuming Z a = 0 for simplicity.Using R = e r e r e r , , we can compute some components of Eq. (3.126) asCYBE a a b b b = − (cid:2) f cd [ b r | c | b b ] r a a d + f cd [ a (cid:0) r a ] cdb b b − r a ][ cd r b b b ] (cid:1)(cid:3) , (5.48)CYBE a a = 2 β f cd [ a r a ] cd = 0 . (5.49)The first equation is precisely the CYBE found in [9] but the second one is weaker than theknown one f cda r bcd = 0 . (5.50)Actually, this condition comes from the non-negative-level part (coefficients of K ab ) of therelation X a a = X a a and thus the conditions (3.126) are not sufficient for Eq. (3.118).When n ≤ 6, we find f ab ··· b = 0 and Eqs. (5.48) and (5.50) will be the only constraintson r a a a and r a ··· a . In n ≤ 7, we obtain further conditions. Here, we do not try to find thefull set of constraints, but consider the CYBE for r a ··· a . For simplicity, we truncate othermulti-vectors and consider R = e r . Then, we have CYBE a a = 0 , and CYBE a a B = 0 andCYBE a ··· a = 0 follow from CYBE a a b b b = − f cd [ a r a ] cdb b b = 0 . The next non-trivialcomponent isCYBE a ··· a b ··· b = − (cid:2) f cd [ b r | c | b ··· b ] r da ··· a + 5 f cd [ a (cid:0) δ a ··· a ] cde ··· e δ [ e f r e ··· e ][ b r b ··· b ] f − r a ··· a ] cd r b ··· b (cid:1)(cid:3) = 0 . (5.51)By using f cd [ a r a ] cdb b b = 0 this is simplified asCYBE a ··· a b ··· b = − (cid:0) f cd [ b r | c | b ··· b ] r da ··· a − f cd [ a r a ··· a ] c [ b r b ··· b ] d (cid:1) = 0 , (5.52)and this can be regarded as the generalized CYBE for r a ··· a . We can also do the samecomputation for r , but we leave further details for future work.37 Exceptional Drinfel’d algebra: type IIB section In this section, we consider the case where the dimension of the subalgebra g is n − T A ) = (cid:0) T a , T a α , T a √ , T a ··· α √ , T a ··· , a √ , T α α ) , T , a α √ , T , a ··· √ , T , a ··· α √ , T , , a (cid:1) , (6.1)where 7 denotes eight totally antisymmetric indices and we also used a shorthand notation a ··· p ≡ a · · · a p . Similar to the case of the M-theory picture, we can check that the dimensionof the EDA coincides with the dimension of the R representation given in Table A.1.Similar to the M-theory case, the embedding tensors for E n ( n ) EDA ( n ≤ 8) in the typeIIB picture are obtained as follows: X a = f abc ˜ K bc + f a βγ R βγ + f ab b β R β b b , + f ab ··· b R b ··· b + f a b ··· b β R β b ··· b + f a b ··· b , b ′ R b ··· b , b ′ − Z a ( ˜ K + t ) , (6.2) X a α = − f bca α ˜ K bc − (cid:0) δ βα f b b a + 2 δ a [ b f b ] αβ (cid:1) R b b β − Z b R ab α , (6.3) X a = − f bca a a ˜ K bc − ǫ βγ f b [ a a β R a ] b γ − f b b [ a R a a ] b b − Z b R a b , (6.4) X a ··· a α = − f bca ··· a α ˜ K bc − f b [ a ··· a R a ] b α + 10 f b [ a a α R a a a ] b − f b b [ a R a ··· a ] b b α − f b αβ R a ··· a b β − Z b R a ··· a b α , (6.5) X a ··· a , a ′ = − (cid:0) f bca ··· a , a ′ − c , f ba ··· a a ′ , c (cid:1) ˜ K bc + ǫ βγ (cid:0) f ba ··· a β R a ′ b γ − c f b [ a ··· a β R a ′ ] b γ (cid:1) − f ba ′ [ a a a R a a a ] b + 35 c f b [ a ··· a R a a a ′ ] b + ǫ βγ (cid:0) f ba ′ [ a β R a ··· a ] b γ − c f b [ a a β R a ··· a a ′ ] b γ (cid:1) − f b b [ a R a ··· a ] b b , a ′ − f bca ′ R a ··· a b , c + 8 Z b R a ··· a b , a ′ − (1 + c , ) Z b R a ··· a a ′ , b , (6.6) X a ··· ( α α ) = − f b [ a ··· a ( α R a ] b α ) + 21 f ba a ( α R a ··· a ] b α ) + f b ( α β ǫ α ) β R a ··· a , b , (6.7) X a ··· , a ′ α = − f ba ··· a , [ a ′ R a ′ ] b α − f b [ a ··· a α R a ] a ′ a ′ b + 21 f ba ′ a ′ [ a a R a ··· a ] b α + 2 f b [ a ′ | b α R a ··· a , | a ′ ] + f ba ′ a ′ α R a ··· a , b , (6.8) X a ··· , a ′ ··· = − f ba ··· a , [ a ′ R a ′ a ′ a ′ ] b − ǫ βγ f b [ a ··· a β R a ] a ′ ··· a ′ b γ + 4 f b [ a ′ a ′ a ′ | b R a ··· a , | a ′ ] + f ba ′ ··· a ′ R a ··· a , b , (6.9) X a ··· , a ′ ··· α = − f ba ··· , [ a ′ R a ′ ··· a ′ ] b α + 6 f b [ a ′ ··· a ′ | b α R a ··· , | a ′ ] + f ba ′ ··· a ′ α R a ··· , b , (6.10) X , , a = f b , b R , a + 2 f b , a R , b , (6.11)where ǫ = ǫ = 1 and constants c , c , c , and c , are defined in Appendix A.2. From thisexpression, we find the vector ϑ A as follows: ϑ a = β f abb − Z a (1 + β δ bb ) , ϑ a α = − β f bba α , ϑ a = − β f bba , ϑ a ··· α = − β f bba ··· α ,ϑ a ··· , a ′ = − β (cid:0) f bba ··· , a ′ − c , f ba ··· a ′ , b (cid:1) , ϑ αβ = ϑ , α = ϑ , = ϑ , α = ϑ , , = 0 . (6.12)38e show a more concise form of the EDA for a particular case, n ≤ T a ◦ T b = f abc T c ,T a ◦ T b β = f acb β T c + f a βγ T b γ − f acb T c β + 2 Z a T b β ,T a ◦ T b b b = f acb b b T c + 3 ǫ γδ f a [ b b γ T b ] δ − f ac [ b T b b ] c + 4 Z a T b b b ,T a ◦ T b ··· b β = f acb ··· b β T c + 5 f a [ b ··· b T b ] β − f a [ b b β T b b b ] + f a βγ T b ··· b γ − f ac [ b T b ··· b ] c β + 6 Z a T b ··· b β ,T a ◦ T b ··· b , b ′ = − ǫ γδ f ab ··· b γ T b ′ δ + 20 f ab ′ [ b b b T b b b ] − ǫ γδ f ab ′ [ b γ T b ··· b ] δ − f ac [ b | T c | b ··· b ] , b ′ − f acb ′ T b ··· b , c + 8 Z a T b ··· b , b ′ ,T a α ◦ T b = f bac α T c + 2 δ a [ b f c ] αγ T c γ + f bca T c α + 4 Z c δ [ ab T c ] α ,T a α ◦ T b β = − f cab α T c β − f c αγ ǫ γβ T cab + ǫ αβ f c c a T c c b − ǫ αβ Z c T abc ,T a α ◦ T b b b = − f ca [ b α T b b ] c − f c αγ T acb b b γ − f c c a T c c b b b α + 2 Z c T ab b b c α ,T a α ◦ T b ··· b β = − f ca [ b α T b ··· b ] c β − ǫ αβ f cda T b ··· b c , d − f c αγ ǫ γβ T b ··· b [ a , c ] + 4 ǫ αβ Z c T ab ··· b , c ,T a α ◦ T b ··· b , b ′ = − f ca [ b | α T c | b ··· b ] , b ′ β − f cab ′ α T b ··· b , c β ,T a a a ◦ T b = − f bca a a T c − ǫ γδ f [ b | [ a a γ δ a ] | c ] T c δ + 3 f bc [ a T a a ] c + 3 f c c [ a δ a b T a ] c c + 16 Z c δ [ a b T a a c ] ,T a a a ◦ T b β = − f ca a a b T c β + 3 f c [ a a β T a ] bc + f c c [ a T a a ] bc c β − Z c T a a a bc β ,T a a a ◦ T b b b = − f ca [ b T b ] c + 3 ǫ γδ f c [ a γ T a ] b c δ + 3 f d [ a δ a c T a ] b d , c + 3 f cd [ a T a ] b c , d + 16 Z c T b [ a , c ] ,T a a a ◦ T b ··· b β = − f ca a a [ b T b ··· b ] c β + 6 f c [ a a β δ a ] cde T b ··· b d , e ,T a a a ◦ T b ··· b , b ′ = − f ca a a [ b | T c | b ··· b ] , b ′ − f ca a a b ′ T b ··· b , c ,T a ··· a α ◦ T b = f ba ··· a c α T c − f [ b [ a ··· a δ a ] c ] T c α − f c [ a a α δ a b T a a ] c + 10 f b [ a a α T a a a ] + 5 f c αγ δ [ a b T a ··· a ] c γ − f b αγ T a ··· a γ + 5 f bc [ a T a ··· a ] c α + 10 f c c [ a δ a b T a a a ] c c α + 36 Z c δ [ a b T a ··· a c ] α ,T a ··· a α ◦ T b β = − f ca ··· a b α T c β − ǫ αβ f c [ a ··· a T a ] bc + 10 f c [ a a α T a a a ] bc β + f c αγ ǫ γβ T ca ··· a , b − ǫ αβ f c c [ a T a ··· a ] c c , b − ǫ αβ Z c T a ··· a c , b ,T a ··· a α ◦ T b b b = − f ca ··· a [ b α T b b ] c + 5 f c [ a ··· a T a ] b b b c α − f c [ a a α δ a a a ] cd d d e T b b b d d d , e ,T a ··· a α ◦ T b ··· b β = − f ca ··· a [ b α T b ··· b ] c β − ǫ αβ f c [ a ··· a δ a ] cde T b ··· b d , e ,T a ··· a ◦ T b ··· b , b ′ = − f ca ··· a [ b | α T c | b ··· b ] , b ′ − f ca ··· a b ′ α T b ··· b , c ,T a ··· a , a ′ ◦ T b = − ǫ γδ f ca ··· a γ δ [ a ′ b T c ] δ − f ba ′ [ a a a T a a a ] + 60 f ca ′ [ a a a δ a b T a a ] c + 6 ǫ γδ f ba ′ [ a γ T a ··· a ] δ − ǫ γδ f ca ′ [ a γ δ a b T a ··· a ] c δ ,T a ··· a , a ′ ◦ T b β = − f ca ··· a β T a ′ bc − f ca ′ [ a a a T a a a ] bc β − f ca ′ [ a β T a ··· a ] c , b ,T a ··· a , a ′ ◦ T b b b = − ǫ γδ f ca ··· a γ T a ′ b b b c δ + 80 f ca ′ [ a a a δ a a a ] cd d d e T b b b d d d , e ,T a ··· a , a ′ ◦ T b ··· b β = − f ca ··· a β T b ··· b [ a ′ , c ] ,T a ··· a , a ′ ◦ T b ··· b , b ′ = 0 . (6.13) n ≤ T a ◦ T b = f abc T c ,T a ◦ T b β = f acb β T c + f a βγ T b γ − f acb T c β + 2 Z a T b β ,T a ◦ T b b b = f acb b b T c + 3 ǫ γδ f a [ b b γ T b ] δ − f ac [ b T b b ] c + 4 Z a T b b b ,T a ◦ T b ··· b β = 5 f a [ b ··· b T b ] β − f a [ b b β T b b b ] + f a βγ T b ··· b γ − f ac [ b T b ··· b ] c β + 6 Z a T b ··· b β ,T a α ◦ T b = f bac α T c + 2 δ a [ b f c ] αγ T c γ + f bca T c α + 4 Z c δ [ ab T c ] α ,T a α ◦ T b β = − f cab α T c β − f c αγ ǫ γβ T cab + ǫ αβ f c c a T c c b − ǫ αβ Z c T abc ,T a α ◦ T b b b = − f ca [ b α T b b ] c − f c αγ T acb b b γ − f c c a T c c b b b α + 2 Z c T ab b b c α ,T a α ◦ T b ··· b β = − f ca [ b α T b ··· b ] c β ,T a a a ◦ T b = − f bca a a T c − ǫ γδ f [ b | [ a a γ δ a ] | c ] T c δ + 3 f bc [ a T a a ] c + 3 f c c [ a δ a b T a ] c c + 16 Z c δ [ a b T a a c ] ,T a a a ◦ T b β = − f ca a a b T c β + 3 f c [ a a β T a ] bc + f c c [ a T a a ] bc c β − Z c T a a a bc β ,T a a a ◦ T b b b = − f ca a a [ b T b b ] c + 3 ǫ γδ f c [ a a γ T a ] b b b c δ ,T a a a ◦ T b ··· b β = − f ca a a [ b T b ··· b ] c β ,T a ··· a α ◦ T b = − f [ b [ a ··· a δ a ] c ] T c α − f c [ a a α δ a b T a a ] c + 10 f b [ a a α T a a a ] + 5 f c αγ δ [ a b T a ··· a ] c γ − f b αγ T a ··· a γ + 5 f bc [ a T a ··· a ] c α + 10 f c c [ a δ a b T a a a ] c c α + 36 Z c δ [ a b T a ··· a c ] α ,T a ··· a α ◦ T b β = − ǫ αβ f c [ a ··· a T a ] bc + 10 f c [ a a α T a a a ] bc β ,T a ··· a α ◦ T b b b = 5 f c [ a ··· a T a ] b b b c α ,T a ··· a α ◦ T b ··· b β = 0 . (6.14)We can obtain the lower-dimensional EDA in a similar manner by a truncation.In the type IIB picture, the level-0 part of the embedding tensor is b Θ (0) a = f abc K bc + f a βγ R βγ − Z a K , (6.15)and the Leibniz identity (3.40) requires f [ abd f c ] de = 0 , f a γβ f b βδ − f b γβ f a βδ + f abc f c γδ = 0 , f abc Z c = 0 . (6.16)Then, defining f ( x ) ≡ x a f ab b β R β b b , f ( x ) ≡ x a f ab ··· b R b ··· b ,f ( x ) ≡ x a f ab ··· b β R β b ··· b , f , ( x ) ≡ x a f ab ··· b , b ′ R b ··· b , b ′ , (6.17)40e can express the cocycle conditions as0 = d f ( x ) , (6.18)0 = d f ( x ) − [ f ( x ) , f ( y )] = 0 , (6.19)0 = d f ( x ) − [ f ( x ) , f ( y )] − [ f ( x ) , f ( y )] , (6.20)0 = d f , ( x ) − [ f ( x ) , f ( y )] − [ f ( x ) , f ( y )] − [ f ( x ) , f ( y )] , (6.21)whered f ∗ ( x, y ) ≡ x · f ∗ ( y ) − y · f ∗ ( x ) − f ∗ ([ x, y ]) , x · f ∗ ( y ) ≡ x a (cid:2) f ∗ ( y ) , b Θ (0) a (cid:3) . (6.22)More explicitly, we find0 = 4 f [ a | d [ c | f | b ] d | c ] γ − f abd f dc c γ − f [ a | γδ f | b ] c c δ − Z [ a f b ] c c γ , (6.23)0 = 8 f [ a | d [ c | f | b ] d | c c c ] − f abd f dc ··· c − ǫ γδ f a [ c c γ f bc c ] δ − Z [ a f b ] c ··· c , (6.24)0 = 12 f [ a | d [ c | f | b ] d | c ··· c ] γ − f abd f dc ··· c γ − f [ a | γδ f | b ] c ··· c δ + 30 f [ a | [ c c α f | b ] c ··· c ] − Z [ a f b ] c ··· c γ , (6.25)0 = 14 f [ a | d [ c | f | b ] d | c ··· c ] , c ′ + 2 f d [ ac ′ f b ] c ··· c , d − f abd f dc ··· c , c ′ − ǫ αβ f [ a | [ c c α f | b ] c ··· c ] c ′ β + 35 f [ a [ c ··· c f b ] c c c ] c ′ − Z [ a f b ] c ··· c , c ′ . (6.26)By introducing r ≡ r a a α R αa a , r ≡ r a ··· a R a ··· a , r ≡ r a ··· a α R αa ··· a , r , ≡ r a ··· a ,a ′ R a ··· a ,a ′ , (6.27)and defining R ≡ e r e r e r e r , , the coboundary ansatzes are decomposed as f ( x ) = d r ( x ) , (6.28) f ( x ) = d r ( x ) + [ r , f ( x )] , (6.29) f ( x ) = d r ( x ) + [ r , f ( x )] − [ r , [ r , f ( x )]] , (6.30) f , ( x ) = d r , ( x ) + [ r , f ( x )] + [ r , f ( x )] − [ r , [ r , f ( x )]] − [ r , [ r , f ( x )]] + [ r , [ r , [ r , f ( x )]]] , (6.31)where d r ∗ ( x ) ≡ x a [ r ∗ , b Θ (0) a ] . In components, we have f ab b β = 2 f ac [ b | r c | b ] β − f a βγ r b b γ − Z a r b b β , (6.32) f ab ··· b = 4 f ac [ b | r c | b b b ] − Z a r b ··· b − ǫ αβ f a [ b b α r b b ] β , (6.33) f ab ··· b β = 6 f ac [ b | r c | b ··· b ] β − f a βγ r b ··· b γ − Z a r b ··· b β − f a [ b ··· b r b b ] β − ǫ γδ f a [ b b γ r b b δ r b b ] α , (6.34)41 ab ··· b , b ′ = 7 f ac [ b | r c | b ··· b ] , b ′ + f acb ′ r b ··· b , c − Z a r b ··· b , b ′ + 21 ǫ γδ r [ b b γ f ab ··· b ] b ′ δ − r [ b ··· b f ab b b ] b ′ + ǫ αβ r b ′ [ b f ab ··· b r b b ] β + ǫ γδ r [ b ··· b (cid:0) r b b γ f ab ] b ′ δ + r b | b ′ | γ f ab b ] δ (cid:1) + ǫ αβ ǫ γδ r [ b b α r | b ′ | b β r b b γ f ab b ] δ . (6.35)For the fundamental identities, assuming n ≤ f cda α f db b β − f da [ b α f cb ] d β = f d αγ ǫ γβ f cab d + ǫ αβ f d a f cb d + 2 ǫ αβ f cab d Z d , (6.36) f cda f eb ··· − f da [ b f cb ] d = − ǫ γδ f d [ a γ f ca ] b ··· d δ − f d [ a δ a e f ca ] b ··· d , e + 3 f d [ a f ca a ] b ··· [ d , d ] + 16 Z d f cb ··· [ a , d ] , (6.37) f cda ··· α f eb ··· β − f da ··· [ b α f cb ··· ] d β = − ǫ αβ f d [ a ··· δ a ] de e f cb ··· [ e , e ] . (6.38)From Z ABC X C = 0, we find additional identities, such as Z a α b C X C = δ ba (cid:0) f dec γ f c αγ + f dec α Z c (cid:1) K de = 0 ⇔ f abc γ f c αγ + f abc α Z c = 0 . (6.39)Again, they are not the whole set of the Leibniz identities.Now, let us construct the generalized frame fields. Using the level-0 generators, we define E AI ≡ (cid:0) e − h αβ R αβ e − ( h R ) ab ˜ K ab (cid:1) AI . (6.40)Here, h R is related to the right-invariant vector fields as e m a = (e − ( h R ) ) a m and we also define λ α ˙ β ≡ (e − h ) α ˙ β . We further multiply e ( ˜ K + t )∆ and Π ≡ e − π a a α R α a a e − π a ··· a R a ··· a e − π a ··· a α R α a ··· a e − π a ··· a , a ′ R a ··· a , a ′ , (6.41)to obtain the generalized frame fields E AI . For example, when n ≤ E a = e a , E a α = π ab α e b + e − λ α ˙ α r a , (6.42) E a a a = (cid:0) π a a a b + ǫ γδ π [ a | b | γ π a a ] δ (cid:1) e b + 3 e − ǫ γδ λ γ ˙ α π [ a a δ r a ] + e − r a a a , (6.43) E a ··· a α = (cid:0) π a ··· a b α − π b [ a a a π a a ] α − ǫ γδ π b [ a γ π a a δ π a a ] α (cid:1) e b − − λ α ˙ α π [ a ··· a r a ] + 15 e − ǫ γδ λ γ ˙ α π [ a a δ π a a α r a ] + 10 e − π [ a a α r a a a ] + e − λ α ˙ α r a ··· a , (6.44) E a ··· , a ′ = (cid:0) ǫ γδ π b [ a ··· a γ π a ] a ′ δ + 30 ǫ γδ π a ′ [ a γ π a a δ π a a a ] b + 10 π b [ a a a π a a a ] a ′ − ǫ αβ ǫ γδ π a ′ [ a α π a a β π a a γ π a ] b δ (cid:1) e b + e − λ β ˙ α (cid:2) − ǫ βγ π a ··· γ r a ′ + 30 (cid:0) ǫ βγ π [ a ··· + ǫ γδ ǫ βǫ π [ a δ π a ǫ (cid:1) π | a ′ | a γ r a ] (cid:3) − 20 e − π a ′ [ a r a ] + 30 e − ǫ γδ π a ′ [ a γ π a δ r a ] + 6 e − λ γ ˙ α ǫ γδ π a ′ [ a δ r a ··· ] + e − r a ··· ⊗ r a ′ , (6.45)where r a ··· a p ≡ r a ∧ · · · ∧ r a p . 42n the general case, the differential identity gives D a π = f a + [ π , b Θ (0) ] , (6.46) D a π = f a + [ π , b Θ (0) ] + [ π , f a ] , (6.47) D a π = f a + [ π , b Θ (0) ] + [ π , f a ] + [ π , [ π , f a ]] , (6.48) D a π , = f a , + [ π , , b Θ (0) ] + [ π , f a ] + [ π , f a ]+ [ π , [ π , f a ]] + [ π , [ π , f a ]] + [ π , [ π , [ π , f a ]] , (6.49) D a ∆ = Z a , ( λ D a λ − ) αβ = f a αβ , (6.50)In components, we find D a π b b β = f ab b β + 2 f ac [ b | π c | b ] β − f a βγ π b b γ − Z a π b b β , (6.51) D a π b ··· b = f ab ··· b + 4 f ac [ b | π c | b b b ] − Z a π b ··· b − ǫ αβ f a [ b b α π b b ] β , (6.52) D a π b ··· b β = f ab ··· b β + 6 f ac [ b | π c | b ··· b ] β − f a βγ π b ··· b γ − Z a π b ··· b β − f a [ b ··· b π b b ] β + 30 ǫ γδ f a [ b b γ π b b δ π b b ] α , (6.53) D a π b ··· , b ′ = f ab ··· , b ′ + 7 f ac [ b | π c | b ··· ] , b ′ + f acb ′ π b ··· , c − Z a π b ··· , b ′ + 21 ǫ γδ f ab ′ [ b ··· γ π b ] δ + f ab ′ [ b π b ··· ] − ǫ αβ f a [ b ··· π b α π b ] b ′ β + ǫ γδ f a [ b γ π b ··· π b ] b ′ δ − ǫ γδ f ab ′ [ b γ π b ··· π b ] δ − ǫ αβ ǫ γδ π [ b α π b | b ′ | β π b γ f ab δ . (6.54)If we define the Nambu–Lie structures as π m m α ≡ π b b α e m b e m b , π m ··· m ≡ π b ··· b e m b · · · e m b ,π m ··· m α ≡ π b ··· b α e m b · · · e m b , π m ··· m ,m ′ ≡ π b ··· b , b ′ e m b · · · e m b e m ′ b ′ , (6.55)we find £ v a π m m α = e − f ab b α v m b v m b , £ a ∆ = Z a , ( λ £ a λ − ) αβ = f a αβ , (6.56) £ v a π m ··· m + 3 ǫ αβ π [ m m α £ v a π m m ] β = e − f ab b v m b v m b , (6.57) £ v a π m ··· α + 15 (cid:0) δ γα π [ m ··· − ǫ βγ π [ m α π m β (cid:1) £ v a π m ] γ = e − f ab ··· v m b · · · v m b , (6.58) £ v a π m ··· ,m ′ − ǫ αβ π m ′ [ m ··· £ v a π m ] − π [ m ··· £ v a π m ] m ′ ǫ αβ π [ m ··· π m α £ v a π m ] m ′ β − ǫ αβ π [ m ··· π m | m ′ | α £ v a π m β − ǫ αβ ǫ γδ π [ m α π m | m ′ | β π m γ £ v a π m δ = e − f ab ··· , b ′ v m b · · · v m b v m ′ b ′ . (6.59)In the case of the coboundary EDA, we can solve for the Nambu–Lie structure by meansof the classical r -matrices. Here, use the formula (5.47) for simplicity. Then we find U I J = (cid:0) e e − ρ e e − ρ e e − ρ e e − ρ , (cid:1) I J . (6.60)43ne can obtain the Nambu–Lie structure Π I J ≡ E I A Π AB E BJ by computing Π I J = R I K (Ω − ) K J , R I J ≡ E I A R AB E BJ . (6.61)For example, we have π m m α = r a a β (cid:0) λ αβ e − v m a v m a − δ βα e m a e m a (cid:1) , (6.62) π m ··· m = r a ··· a (cid:0) e − v m a · · · v m a − e m a · · · e m a (cid:1) − − ǫ γδ π [ m m r m m ] , (6.63)which are similar to the relations (5.40)–(5.42) found in the M-theory picture.In the fully general situation, for example, we obtain the generalized CYBE for π a a α asCYBE a α b b β = − (cid:2) f cd [ b r b ] d β r ca α − f c βγ r ac α r b b γ − Z c r ac α r b b β + (cid:0) f cda − Z [ c δ ad ] (cid:1)(cid:0) ǫ αβ r cdb b − r c [ b ( α r b ] d β ) − r cd [ α r b b β ] (cid:1) + f c αγ (cid:0) ǫ γβ r acb b − r a [ b ( γ r b ] c β ) − r ac [ γ r b b β ] (cid:1)(cid:3) = 0 . (6.64)By considering CYBE (3.118) for non-negative generators and CYBE aα = 0, we find additionalconditions, such as r ab α Z b = 0 , r bc α f bca = 0 , r ab α f b βγ = 0 . (6.65)In particular, when ( r ab α ) = ( r ab , r ab ) = ( r ab , 0) and f a = 0 are satisfied, we findCYBE a b b = 3 f cd [ a r | c | b r b ] d + 3 r [ ab r b ] c ( f c + 2 Z c (cid:1) = 0 , (6.66)which reduces to the standard classical Yang–Baxter equation when the dilaton flux satisfies f a = − Z a . The second requirement of Eq. (6.65) shows that the classical matrix shouldbe unimodular in the terminology of [66].We can straightforwardly compute other components of the CYBE, but this is a veryhard task. Here, assuming f a βγ = 0 and Z a = 0, we show the main part of the CYBE formulti-vectors. For simplicity, again we consider the case of a single multi-vector. Since r a a α is already considered, let us consider R = e r and R = e r . In each case, we findCYBE a b ··· = − (cid:2) f cd [ b r | c | b ] r da + 3 f cd [ a (cid:0) δ a ] cde ··· δ [ e f r e ][ b r b ] f − r a ] cd r b ··· (cid:1)(cid:3) = 0 , (6.67)CYBE a ··· α b ··· β = − (cid:2) f cd [ b r | c | b ] α r da β − f cd [ a δ a ··· ] cde ··· e δ [ e f r e ··· ][ b α r b ··· ] f β − f cd [ a (cid:0) r a ··· ] cd β r b ··· α − r a ··· ] cd α r b ··· β (cid:1)(cid:3) = 0 . (6.68) From f a ββ = 0 , the dilaton flux satisfies f a = − f a . Summary and discussions In this paper, we presented the general structure of the ExDA by using the generalized Liederivative in ExFT. As is known in the Poisson–Lie T -duality, if we choose a maximallyisotropic subalgebra g of the ExDA, we can systematically construct the generalized framefields E AI . We have generally shown that, for any choice of the subalgebra g , the constructedgeneralized frame fields E AI satisfy the algebraˆ £ E A E BI = − X ABC E C I , (7.1)which means that the target space can be called a generalized parallelizable space. Here, wehave considered DFT and the E n ( n ) EFT ( n ≤ 8) as particular examples of ExFT, but ourpresentation does not depend on the details of the duality group G . Thus the generalizedparallelizability will be realized even for other ExFTs, such as the heterotic DFT [67]. It isalso interesting to consider the extension to the E EFT (see for example [68]).Among the Leibniz identities, the particularly interesting ones are the cocycle conditions.By using the duality algebra, we found a general definition of the coboundary operator δ n .In particular, the operator δ is non-trivial and the coboundary ansatz (3.111) shows that theembedding tensor b Θ a is twisted by the classical r -matrices. Moreover, we have provided ageneral formula (6.61) of the Nambu–Lie structure π for a general coboundary ExDA.As a particular class of the non-Abelian duality, we discussed the Yang–Baxter deforma-tions. The explicit form of the generalized CYBE needs to be clarified in future studies, butonce a solution of the CYBE is found, we can easily perform the Yang–Baxter deformation, M IJ → M ′ IJ = (cid:0) Ω ∆ M Ω T ∆ (cid:1) IJ . (7.2)Here, the matrix (Ω ∆ ) I J is made of the multi-vector fields e − ( p + q )∆ ρ i ··· i p ,i p +1 ··· i p + q where ρ i ··· i p ,i p +1 ··· i p + q = r a ··· a p ,a p +1 ··· a p + q v i a · · · v i p + q a p + q , (7.3)and r is the generalized classical r -matrix. Recently, the Yang–Baxter deformations have beenunderstood as the local β -transformations [69–75] which are characterized by the bi-vector ρ mn = r ab v ma v nb . As a natural extension, the tri-vector deformation in 11D supergravity hasbeen proposed in [76, 77], and our general formula (7.2) supports the conjectural multi-vectordeformations. In this paper, we consider that the physical space is a group manifold, butthe usual Yang–Baxter deformation can be performed in more general curved spaces. It isimportant future work to extend the generalized Yang–Baxter deformation to a more generalsetup. It is also important to study the non-Abelian duality for non-coboundary ExDAs. When we consider concrete examples, we can easily check the CYBE by computing X ABC = X ABC (recallEq. (3.118)). The only difficulty is to identity the full set of independent conditions on a general r A from X ABC = X ABC . 45n this paper, we have parameterized the generators of g as T a and introduced the corre-sponding physical coordinates x a (= δ ai x i ) . However, we can change this parameterization.For example, in the M-theory picture, we may find a maximally isotropic subalgebra g thatis spanned by a mixture of T a and T a a . In this case, we construct the group element g byexponentiating these generators with the corresponding coordinates x a and y a a . In fact,even in this case, we can systematically construct the generalized frame fields by following thesame procedure. By construction, ExFT has the formal duality symmetry, x I → (Λ − ) J I x J , M IJ → Λ I K Λ J L M KL (Λ ∈ G ) , (7.4)where Λ I J is a constant matrix. Then, for an arbitrary Leibniz algebra which is generated bythe generalized Lie derivative and contains a maximally isotropic algebra g , we can performthe formal duality transformation (7.4) such that g is spanned by T ′ a . After the dualityrotation, the Leibniz algebra will have the form of an ExDA. In other words, if a Leibnizalgebra generated by the generalized Lie derivative contains a maximally isotropic algebra, itwill be related to an ExDA through a formal duality. In this sense, EDA is very universal.In this paper, we have defined the ExDA by using the generalized Lie derivative in ExFT. Inthis case, the structure constants have the form (1.22), which has been studied in the maximalgauged supergravities. If we consider less supersymmetric cases, the structure constants willhave a more general form. It is an interesting to study whether we can construct certaingeneralized parallelizable spaces for this generalized algebras. For example, in the case of theusual Drinfel’d double, a deformation of the structure constants X ABC , X ABC → X ′ ABC ≡ X ABC + 2 X [ A δ CB ] , (7.5)was studied in [78, 79]. Here, the vector X A is supposed to be a null vector η AB X A X B = 0 .To be more explicit, by parameterizing ( X A ) = ( α a , β a ) and defining f ′ abc ≡ f abc + 2 α [ a δ cb ] and f ′ cab ≡ f cab + 2 β [ a δ c ] b , this deformed Lie algebra is given by T a ◦ T b = f ′ abc T c , T a ◦ T b = f ′ cab T c ,T a ◦ T b = (cid:0) f ′ abc + β c δ ba − β b δ ca (cid:1) T c − (cid:0) f ′ acb − α c δ ba + 2 α a δ bc (cid:1) T c = − T b ◦ T a . (7.6)In [78, 79], the standard procedures of the Poisson–Lie T -duality was straightforwardly ex-tended and the generalized duality, called the Jacobi–Lie T -duality, was discussed. Obviously,we find X ′ [ ABC ] = X ′ ABC and we cannot realize this algebra by means of the generalized Liederivative in DFT. Deformations of the generalized Lie derivative in ExFT have been studiedin [80–82] and it is interesting to extend the ExDA by using such deformed Lie derivatives. Acknowledgments This work is supported by JSPS Grant-in-Aids for Scientific Research (C) 18K13540 and (B)18H01214. 46 Conventions We (anti-)symmetrize the indices as A ( a ··· a p ) = p ! (cid:2) A a ··· a p + (permutations) (cid:3) , A [ a ··· a p ] = p ! (cid:2) A a ··· a p ± (permutations) (cid:3) . (A.1)We define the antisymmetrized Kronecker delta as δ a ··· a p b ··· b p ≡ δ [ a [ b · · · δ a p ] b ] .We sometimes use a short-hand notation A a ··· p ≡ A a ··· p and may simply denote it as A p .In this Appendix, we use a further condensed notation, such as A ¯ a p ¯ b q ≡ √ p ! q ! A a ··· p b ··· q . (A.2)Namely, in order to reproduce the standard notation, each barred index ¯ a p is replaced by theantisymmetrized indices a ··· p and we additionally multiply the factor √ p ! to the expression. Inthis notation, the antisymmetrized Kronecker delta δ a ··· a p b ··· b p appears with a factor p ! . Namely,for example, we have δ ¯ a p ¯ b p ≡ √ p ! p ! p ! δ a ··· p b ··· p = δ a ··· p b ··· p , δ c ¯ a p ¯ b p +1 ≡ √ p ! ( p +1)! ( p + 1)! δ ca ··· p b ··· ( p +1) . (A.3)This notation significantly simplifies various expression.For E n ( n ) group, we define various representations as described in Table A.1. n R R R R R R R R R R SC 21 2 1 1 2 21 31 32 4 × , ) ( , ) ( , ) ( , ) ( , ) ( , )( , ) ( , )( , ) ( , )( , )2 × ( , ) ( , )4 10 5 5 10 24 4015 70455 5 16 10 16 45 144 32012610 10 27 27 78 351 172827 27 56 133 912 8645133 1331 248 38751 1472503875248 38752481 Table A.1: Representations of E n ( n ) group. The R p representation is the U -duality multipletof p -form fields in (11 − n ) dimensions [83, 84]. The adjoint representation R adj of E n ( n ) coincides with R − n and the embedding tensor Θ A α transforms in the R − n representation.47 .1 E n ( n ) algebra in the M-theory picture In the M-theory picture, the non-vanishing commutators for the e n ( n ) algebra ( n ≤ 8) aregiven as follows:[ K ab , K cd ] = δ cb K ad − δ ad K cb , (A.4)[ K ab , R ¯ c ] = δ ¯ c b ¯ d R a ¯ d , [ K ab , R ¯ c ] = − δ a ¯ d ¯ c R b ¯ d , (A.5)[ K ab , R ¯ c ] = δ ¯ c b ¯ d R a ¯ d , [ K ab , R ¯ c ] = − δ a ¯ d ¯ c R b ¯ d , (A.6)[ K ab , R ¯ c ,c ′ ] = δ ¯ c b ¯ d R a ¯ d ,c ′ + δ c ′ b R ¯ c ,a , [ K ab , R ¯ c ,c ′ ] = − δ a ¯ d ¯ c R b ¯ d ,c ′ − δ ac ′ R ¯ c ,b , (A.7)[ R ¯ a , R ¯ b ] = R ¯ a ¯ b , [ R ¯ a , R ¯ b ] = δ ¯ b ¯ c c ′ R ¯ a ¯ c ,c ′ , (A.8)[ R ¯ a , R ¯ b ] = − δ ¯ a ¯ e c δ ¯ e d ¯ b K cd + δ ¯ a ¯ b K , [ R ¯ a , R ¯ b ] = − δ ¯ a ¯ c ¯ b R ¯ c , (A.9)[ R ¯ a , R ¯ b ,b ] = − δ ¯ a ¯ c ¯ b R ¯ c b , [ R ¯ a , R ¯ b ] = δ ¯ a ¯ b ¯ c R ¯ c , (A.10)[ R ¯ a , R ¯ b ] = − δ ¯ a ¯ e c δ ¯ e d ¯ b K cd + δ ¯ a ¯ b K , [ R ¯ a , R ¯ b ,b ] = − δ ¯ a ¯ c ¯ b R ¯ c b , (A.11)[ R ¯ a ,a , R ¯ b ] = δ ¯ a ¯ b ¯ c R ¯ c a , [ R ¯ a ,a , R ¯ b ] = δ ¯ a ¯ b ¯ c R ¯ c a , (A.12)[ R ¯ a ,a , R ¯ b ,b ] = − δ ¯ a ¯ b K ab , [ R ¯ a , R ¯ b ] = R ¯ a ¯ b , [ R ¯ a , R ¯ b ] = δ ¯ c c ′ ¯ b R ¯ a ¯ c ,c ′ , (A.13)where K ≡ K aa .The matrix representations ( t α ) AB in the R representation are as follows: K cd ≡ ˜ K cd − β n δ cd t , (A.14)˜ K cd ≡ δ ca δ bd K K K , K , K , 00 0 0 0 0 0 K , , (A.15) K p ≡ − δ ¯ a p ¯ d ¯ e p − δ ¯ c ¯ e p − ¯ b p , K s,t ≡ − δ ¯ a s d ¯ e s − δ c ¯ e s − ¯ b s δ ¯ a ′ t ¯ b ′ t − δ ¯ a s ¯ b s δ ¯ a ′ t d ¯ e t − δ c ¯ e t − ¯ b ′ t ,K , , ≡ − δ ¯ a d ¯ e δ c ¯ e ¯ b δ ¯ a ′ ¯ b ′ δ a ′′ b ′′ − δ ¯ a ¯ b δ ¯ a ′ d ¯ e δ c ¯ e ¯ b ′ δ a ′′ b ′′ − δ ¯ a ¯ b δ ¯ a ′ ¯ b δ a ′′ d δ cb ′′ , (A.16) R ¯ c ≡ δ b ¯ a ¯ c − δ ¯ a ¯ b ¯ c − δ ¯ a ¯ b ¯ d δ ¯ d a ′ ¯ c + δ ¯ a a ′ ¯ b ¯ c − δ ¯ a ¯ b d δ d ¯ e ¯ c δ ¯ a ′ ¯ e b ′ + δ ¯ a ¯ b b ′ δ ¯ a ′ ¯ c − δ ¯ a ¯ b δ ¯ a ′ ¯ b ′ ¯ c − δ ¯ a ¯ b δ ¯ a ′ ¯ b ′ ¯ d δ a ′′ ¯ d ¯ c , (A.17)48 ¯ c ≡ − δ ¯ c a ¯ b δ ¯ a ¯ c ¯ b δ ¯ a ¯ d ¯ b δ ¯ c ¯ d b ′ − δ ¯ a ¯ c ¯ b b ′ δ ¯ a d ¯ b δ ¯ c d ¯ e δ ¯ e a ′ ¯ b ′ − δ ¯ a a ′ ¯ b δ ¯ c ¯ b ′ δ ¯ a ¯ b δ ¯ a ′ ¯ c ¯ b ′ 00 0 0 0 0 0 δ ¯ a ¯ b δ ¯ a ′ ¯ d ¯ b ′ δ ¯ c b ′′ ¯ d , (A.18) R ¯ c ≡ δ b ¯ a ¯ c − δ a ′ d ¯ b δ ¯ a ¯ c d − δ ¯ a a ′ ¯ c ¯ b − δ ¯ a ¯ b ¯ d δ ¯ a ′ ¯ d ¯ c − δ ¯ a d ¯ b δ ¯ a ′ d ¯ c b ′ − δ ¯ a ¯ b b ′ δ ¯ a ′ ¯ c δ ¯ a ¯ b δ ¯ a ′ ¯ c ¯ d δ a ′′ ¯ d ¯ b ′ , (A.19) R ¯ c ≡ − δ ¯ c a ¯ b δ ¯ a b ′ d δ ¯ c d ¯ b + δ ¯ c ¯ a ¯ b b ′ δ ¯ a ¯ d ¯ b δ ¯ c ¯ b ′ ¯ d δ d ¯ a ¯ b δ ¯ c a ′ ¯ b ′ d + δ ¯ a a ′ ¯ b δ ¯ c ¯ b ′ 00 0 0 0 0 0 − δ ¯ a ¯ b δ ¯ c ¯ d ¯ b ′ δ ¯ a ′ b ′′ ¯ d , (A.20) R ¯ c ,c ′ ≡ δ b ¯ a ¯ c δ a ′ c ′ − δ ¯ a a ′ ¯ c δ bc ′ δ ¯ a ¯ c δ ¯ a ′ c ′ ¯ b δ ¯ a ¯ c δ ¯ a ′ c ′ ¯ b δ ¯ a ¯ c δ ¯ a ′ c ′ ¯ b δ a ′′ b ′ − δ ¯ a ¯ c δ ¯ a ′ ¯ b b ′ δ a ′′ c ′ , (A.21) R ¯ c ,c ′ ≡ − δ ¯ c a ¯ b δ c ′ b ′ + δ ¯ c ¯ b b ′ δ c ′ a − δ ¯ c ¯ b δ c ′ ¯ a ¯ b ′ − δ ¯ c ¯ b δ c ′ ¯ a ¯ b ′ 00 0 0 0 0 0 − δ ¯ c ¯ b δ c ′ ¯ a ¯ b ′ δ a ′ b ′′ + δ ¯ c ¯ b δ ¯ a a ′ ¯ b ′ δ c ′ b ′′ . (A.22)49 .2 E n ( n ) algebra in the type IIB picture In the type IIB picture, the non-vanishing commutators for the e n ( n ) algebra ( n ≤ 8) are givenas follows:[ K ab , K cd ] = δ cb K ad − δ ad K cb , (A.23)[ K ab , R ¯ c α ] = δ ¯ c bd R ad α , [ K ab , R α ¯ c ] = − δ ad ¯ c R α bd , (A.24)[ K ab , R ¯ c ] = δ ¯ c b ¯ d R a ¯ d , [ K ab , R ¯ c ] = − δ a ¯ d ¯ c R b ¯ d , (A.25)[ K ab , R ¯ c α ] = δ ¯ c b ¯ d R a ¯ d α , [ K ab , R α ¯ c ] = − δ a ¯ d ¯ c R α b ¯ d , (A.26)[ K ab , R ¯ c , c ′ ] = δ ¯ c b ¯ d R a ¯ d , c ′ + δ c ′ b R ¯ c , a , [ K ab , R ¯ c , c ′ ] = − δ a ¯ d ¯ c R b ¯ d , c ′ − δ ac ′ R ¯ c , b , (A.27)[ R αβ , R γδ ] = δ γβ R αδ − δ αδ R γβ , (A.28)[ R αβ , R ¯ c γ ] = − (cid:0) δ αγ δ δβ − δ αβ δ δγ (cid:1) R ¯ c δ , [ R αβ , R γ ¯ c ] = (cid:0) δ αδ δ γβ − δ αβ δ γδ (cid:1) R δ ¯ c , (A.29)[ R αβ , R ¯ c γ ] = − (cid:0) δ αγ δ δβ − δ αβ δ δγ (cid:1) R ¯ c δ , [ R αβ , R γ ¯ c ] = (cid:0) δ αδ δ γβ − δ αβ δ γδ (cid:1) R δ ¯ c , (A.30)[ R ¯ a α , R ¯ b β ] = ǫ αβ R ¯ a ¯ b , [ R ¯ a α , R ¯ b ] = − R ¯ a ¯ b α , [ R ¯ a α , R ¯ b β ] = ǫ αβ δ ¯ b ¯ c f R ¯ a ¯ c , f , (A.31)[ R ¯ a α , R β ¯ b ] = − δ βα δ ¯ a ce δ de ¯ b K cd + δ βα δ ¯ a ¯ b K + δ ¯ a ¯ b R βα , (A.32)[ R ¯ a α , R ¯ b ] = − ǫ αβ δ ¯ a ¯ c ¯ b R β ¯ c , [ R ¯ a α , R β ¯ b ] = δ βα δ ¯ a ¯ c ¯ b R ¯ c , (A.33)[ R ¯ a α , R ¯ b , b ′ ] = − ǫ αβ δ ¯ a ¯ c ¯ b R β ¯ c b ′ , (A.34)[ R ¯ a , R ¯ b ] = − δ ¯ b ¯ c d R ¯ a ¯ c , d , [ R ¯ a , R α ¯ b ] = ǫ αβ δ ¯ a ¯ b ¯ c R ¯ c β , (A.35)[ R ¯ a , R ¯ b ] = − δ ¯ a c ¯ e δ d ¯ e ¯ b K cd + δ ¯ a ¯ b K (A.36)[ R ¯ a , R α ¯ b ] = − δ ¯ a ¯ c ¯ b R α ¯ c , [ R ¯ a , R ¯ b , b ′ ] = δ ¯ a ¯ c ¯ b R ¯ c b ′ (A.37)[ R ¯ a α , R β ¯ b ] = − δ βα δ ¯ a ¯ b ¯ c R ¯ c , [ R ¯ a α , R ¯ b ] = δ ¯ a ¯ b ¯ c R ¯ c α , (A.38)[ R ¯ a α , R β ¯ b ] = − δ βα δ ¯ a c ¯ e δ d ¯ e ¯ b K cd + δ βα δ ¯ a ¯ b K + δ ¯ a ¯ b R βα , (A.39)[ R ¯ a α , R ¯ b , b ] = ǫ αβ δ ¯ a c ¯ b R β cb , (A.40)[ R ¯ a , a ′ , R α ¯ b ] = ǫ αβ δ ¯ a ¯ b ¯ c R ¯ c a ′ β , [ R ¯ a , a ′ , R ¯ b ] = − δ ¯ a ¯ b ¯ c R ¯ c a ′ (A.41)[ R ¯ a , a , R α ¯ b ] = − ǫ αβ δ ¯ a ¯ b c R ca β , [ R ¯ a , a , R ¯ b , b ] = − δ ¯ a ¯ b K ab , (A.42)[ R α ¯ a , R β ¯ b ] = ǫ αβ R ¯ a ¯ b , [ R α ¯ a , R ¯ b ] = − R α ¯ a ¯ b , [ R α ¯ a , R β ¯ b ] = ǫ αβ δ ¯ c d ¯ b R ¯ a ¯ c , d , (A.43)[ R ¯ a , R ¯ b ] = − δ ¯ c f ¯ b R ¯ a ¯ c , f , (A.44)where K ≡ K aa and ǫ = ǫ = 1 . 50he matrix representation in the R representation are as follows: K cd ≡ ˜ K cd − β n δ cd t , (A.45) K cd ≡ δ ca δ bd δ βα K K δ βα K K , δ ( α ( β δ α ) β ) K δ βα K , K , δ βα K , 00 0 0 0 0 0 0 0 0 K , , , (A.46) K p ≡ − δ ¯ a p d ¯ e p − δ c ¯ e p − ¯ b p , K s,t ≡ − δ c ¯ e s − ¯ b s δ ¯ a s d ¯ e s − δ ¯ a ′ t ¯ b ′ t − δ ¯ a s ¯ b s δ ¯ a ′ t d ¯ d t − δ c ¯ d t − ¯ b ′ t ,K , , ≡ − δ c ¯ e ¯ a δ ¯ b d ¯ e δ ¯ b ′ ¯ a ′ δ b ′′ a ′′ − δ ¯ b ¯ a δ c ¯ e ¯ a ′ δ ¯ b ′ d ¯ e δ b ′′ a ′′ − δ ¯ b ¯ a δ ¯ b ′ ¯ a ′ δ ca ′′ δ b ′′ d , (A.47) R γ δ ≡ R δ ab R δ ¯ a ¯ b R δ ¯ a ¯ b R δ ¯ a ¯ b δ ¯ a ′ ¯ b ′ R δ ¯ a ¯ b δ ¯ a ′ ¯ b ′ 00 0 0 0 0 0 0 0 0 0 , (A.48) h R ≡ δ γα δ βδ − δ βα δ γδ , R ≡ δ γ ( α δ ǫα ) δ ( β δ δ β ) ǫ − δ ( β ( α δ β ) α ) δ γδ i , (A.49) R γ ¯ c ≡ − δ γα δ ab ¯ c − ǫ βγ δ ¯ a b ¯ c − δ γα δ ¯ a ¯ b ¯ c ǫ βγ (cid:20) − δ ¯ a b d δ a ′ d ¯ c − c δ ¯ a a ′ ¯ b c (cid:21) − δ β ( α δ γα ) δ ¯ a ¯ b ¯ c δ γα (cid:20) − δ ¯ a b d δ de ¯ c δ ¯ a ′ eb ′ − c δ ¯ a b b ′ δ ¯ a ′ c (cid:21) − δ ( β α ǫ β ) γ δ ¯ a ¯ b δ ¯ a ′ ¯ c − ǫ βγ δ ¯ a ¯ b δ ¯ a ′ ¯ b ′ ¯ c − δ γα δ ¯ a ¯ b δ ¯ a ′ ¯ b ′ ¯ c − ǫ βγ δ ¯ a ¯ b δ ¯ a ′ ¯ b ′ d δ a ′′ d ¯ c (cid:2) c ≡ + √ (cid:3) , (A.50)51 ¯ c γ ≡ δ βγ δ ¯ c ba ǫ αγ δ a ¯ c ¯ b δ βγ δ ¯ a ¯ c ¯ b ǫ αγ (cid:20) δ ¯ a d ¯ b δ ¯ c b ′ d + c δ ¯ a c b b ′ (cid:21) δ ( β ( α δ β ) γ ) δ ¯ a ¯ c ¯ b δ βγ (cid:20) δ ¯ a d ¯ b δ ¯ c de δ ea ′ ¯ b ′ + c δ ¯ a a ′ ¯ b δ ¯ c b ′ (cid:21) δ β ( α ǫ α ) γ δ ¯ a ¯ b δ ¯ c ¯ b ′ ǫ αγ δ ¯ a ¯ b δ ¯ a ′ ¯ c ¯ b ′ δ βγ δ ¯ a ¯ b δ ¯ a ′ ¯ c ¯ b ′ 00 0 0 0 0 0 0 0 0 ǫ αγ δ ¯ a ¯ b δ ¯ a ′ d ¯ b ′ δ ¯ c b ′′ d , (A.51) R ¯ c ≡ − δ ¯ a b ¯ c δ βα δ ¯ a b ¯ c − (cid:20) δ ¯ a b d δ a ′ ¯ d c + c δ ¯ a a ′ ¯ b c (cid:21) − δ βα δ ¯ a ¯ b ¯ d δ ¯ d ¯ a ′ ¯ c − (cid:20) δ ¯ a b d δ ¯ a ′ e b ′ δ d ¯ e c + c δ ¯ a b b ′ δ ¯ a ′ c (cid:21) δ βα δ ¯ a ¯ b δ ¯ a ′ ¯ b ′ ¯ c − δ ¯ a ¯ b δ a ′ ¯ d ¯ b ′ δ ¯ a ′ ¯ d ¯ c (cid:2) c ≡ + √ (cid:3) , (A.52) R ¯ c ≡ δ ¯ c ¯ b a − δ βα δ a ¯ c ¯ b (cid:20) δ ¯ a d b δ ¯ c b ′ ¯ d + c δ ¯ a c b b ′ (cid:21) δ βα δ ¯ a ¯ d ¯ b δ ¯ c ¯ d ¯ b ′ (cid:20) δ ¯ a d ¯ b δ ¯ e a ′ ¯ b ′ δ ¯ c d ¯ e + c δ ¯ a a ′ ¯ b δ ¯ c b ′ (cid:21) − δ βα δ ¯ a ¯ b δ ¯ a ′ ¯ c ¯ b ′ 00 0 0 0 0 0 0 0 0 δ ¯ a ¯ b δ ¯ a ′ b ′ ¯ d δ ¯ d ¯ c ¯ b ′ , (A.53)52 γ ¯ c ≡ − δ γα δ ¯ a b ¯ c ǫ βγ (cid:20) δ ¯ a c δ a ′ b − c δ ¯ a a ′ b ¯ c (cid:21) δ ( β ( α δ γ ) α ) δ ¯ a b ¯ c δ γα δ ¯ a ¯ c d δ ¯ a ′ d ¯ b − ǫ βγ δ ¯ a ¯ c d δ ¯ a ′ d ¯ b δ γα (cid:20) − δ ¯ a ′ b δ ¯ a b ′ ¯ c + c δ ¯ a b b ′ δ ¯ a ′ c (cid:21) − δ ( β α ǫ β ) γ δ ¯ a ¯ b δ ¯ a ′ ¯ c − ǫ βγ δ ¯ a ¯ b δ ¯ a ′ ¯ c d δ a ′′ d ¯ b ′ , (A.54) R ¯ c γ ≡ δ βγ δ ¯ c ¯ b a ǫ αγ (cid:20) − δ ¯ c b δ ab ′ + c δ a ¯ c b b ′ (cid:21) − δ ( β ( α δ β ) γ ) δ a ¯ c ¯ b − δ βγ δ ¯ c d ¯ b δ ¯ a ¯ b ′ d ǫ αγ δ ¯ c d ¯ b δ ¯ a ¯ b ′ d δ βγ (cid:20) δ ¯ a b ′ δ a ′ ¯ c b − c δ ¯ a a ′ ¯ b δ ¯ c b ′ (cid:21) 00 0 0 0 0 0 0 0 δ β ( α ǫ α ) γ δ ¯ a ¯ b δ ¯ c ¯ b ′ 00 0 0 0 0 0 0 0 0 ǫ αγ δ ¯ a ¯ b δ ¯ c d ¯ b ′ δ ¯ a ′ b ′′ d , (A.55) R ¯ c , c ′ ≡ h δ b ¯ a c δ a ′ c ′ − c , δ ¯ a a ′ ¯ c δ bc ′ i − δ βα δ ¯ a ¯ c δ ¯ a ′ bc ′ − δ ¯ a ¯ c δ ¯ a ′ ¯ b c ′ − δ βα δ ¯ a ¯ c δ ¯ a ′ ¯ b c ′ (cid:20) δ ¯ a c δ ¯ a ′ b c ′ δ a ′′ b ′ − c , δ ¯ a c δ ¯ a ′ b b ′ δ a ′′ c ′ (cid:21) , (A.56) R ¯ c , c ′ ≡ (cid:20) − δ ¯ c a ¯ b δ c ′ b ′ + c , δ ¯ c b b ′ δ c ′ a (cid:21) δ βα δ ¯ c ¯ b δ ac ′ ¯ b ′ δ ¯ c ¯ b δ ¯ a c ′ ¯ b ′ δ βα δ ¯ c ¯ b δ ¯ a c ′ ¯ b ′ 00 0 0 0 0 0 0 0 0 (cid:20) − δ ¯ c b δ ¯ a c ′ ¯ b ′ δ a ′ b ′′ + c , δ ¯ c b δ ¯ a a ′ ¯ b ′ δ c ′ b ′′ (cid:21) h c ≡ − √ c , ≡ + √ i . (A.57)53 .3 Explicit matrix form of χ We here determine the explicit form of the matrices χ α B and χ A β by requiring( t α ) AB = f αβγ χ γ A χ B β . (A.58)In the M-theory picture, the adjoint index α and the vector index B are decomposed as( χ α B ) = (cid:0) χ α b , χ α b √ , χ α b ··· √ , χ α b ··· ,b √ , χ α ,b √ , χ α ,b ··· √ , χ α , ,b (cid:1) , ( χ α B ) = (cid:0) χ R ,a B , χ Ra ··· B √ , χ Ra B √ , χ K ac B , χ Ra B √ , χ Ra ··· B √ , χ R ,a B (cid:1) . (A.59)Under this decomposition, components of ( χ α B ) and ( χ A β ) are determined as follows:( χ α B ) = δ ba δ b ··· a ··· 00 0 0 0 δ b a ǫ ab ··· δ bc − ǫ b ··· b δ ac √ − ǫ a b ··· √ 3! 5! ǫ a ··· b √ 6! 2! δ ab . (A.60)( χ A β ) = δ ab ǫ a b ··· √ 2! 6! 00 0 0 0 ǫ a ··· b √ 5! 3! ǫ ba ··· δ da ′ + ǫ a ··· a ′ δ db √ δ b a δ b ··· a ··· δ ba , (A.61)where ǫ ··· = ǫ ··· = 1 .In the type IIB picture, we consider the following decomposition:( χ α B ) = (cid:0) χ α b , χ α b β , χ α b √ , χ α b ··· β √ , χ α b ··· , b √ , χ α β (12) , χ α , b β √ , χ α , b ··· √ , χ α , b ··· β √ , χ α , , b (cid:1) , ( χ α B ) = (cid:0) χ R , a B , χ Rα a ··· B √ , χ R a ··· B √ , χ Rα a B √ , χ R α α B , χ K cd B , χ R a α B √ , χ R a ··· B √ , χ R a ··· α B √ , χ R , a B (cid:1) . (A.62)Then, the matrices ( χ α B ) and ( χ A β ) are determined as( χ α B ) = δ ba δ αβ δ b ··· a ··· 00 0 0 0 0 0 0 δ b ··· a ··· δ αβ δ b a δ α β ǫ β α ǫ ab ··· δ b ′ c − c , ǫ b ··· b δ ac √ − ǫαβ ǫ a b ··· √ 2! 5! ǫ a ··· b √ 4! 3! ǫαβ ǫ a ··· b √ δ ab . 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