Jacobi-Lie T-plurality

We propose a Leibniz algebra, to be called DD$^+$, which is a generalization of the Drinfel'd double. We find that there is a one-to-one correspondence between a DD$^+$ and a Jacobi--Lie bialgebra, extending the known correspondence between a Lie bialgebra and a Drinfel'd double. We then construct generalized frame fields $E_A{}^M\in\text{O}(D,D)\times\mathbb{R}^+$ satisfying the algebra $\mathcal{L}_{E_A}E_B = - X_{AB}{}^C\,E_C\,$, where $X_{AB}{}^C$ are the structure constants of the DD$^+$ and $\mathcal{L}$ is the generalized Lie derivative in double field theory. Using the generalized frame fields, we propose the Jacobi-Lie T-plurality and show that it is a symmetry of double field theory. We present several examples of the Jacobi-Lie T-plurality with or without Ramond-Ramond fields and the spectator fields.

Unlike the Drinfel'd double, the structure constants X AB C of EDA do not necessarily have the antisymmetry, X AB C = −X BA C , and it is a Leibniz algebra rather than a Lie algebra.
In this paper, we study a minimal extension of the Drinfel'd double by allowing the structure constants to admit the symmetric part X (AB) C = 0 . Using this new Leibniz algebra, we study an extension of the Poisson-Lie T -duality, which we call the Jacobi-Lie T -plurality. 1 The proposed Leibniz algebra has the form and two subalgebras g andg are maximally isotropic with respect to this. If Z a = Z a = 0 , this algebra reduces to the Lie algebra of the Drinfel'd double, but otherwise the "adjointinvariance" is relaxed as follows by allowing for a scale transformation: where T A ≡ (T a , T a ) (A = 1, . . . , 2D) and Z A ≡ (Z a , Z a ) . Since this Leibniz algebra is an extension of the Drinfel'd double by admitting the scale symmetry R + , we call this an extended Drinfel'd algebra DD + . It turns out that this R + symmetry provides a scale factor similar to the trombone symmetry in supergravity [19].
In this paper, we show that the DD + provides an alternative way to define the Jacobi-Lie algebra, and explain how to construct geometric objects such as the Jacobi-Lie structures 1 The Jacobi-Lie T -duality studied in [17,18] is very similar to our proposal, and this paper is strongly inspired by these papers. However, our identification of the supergravity fields is different from the one given in [17,18]. The details are explained in sections 3 and 4. from a given DD + . We also show that we can systematically construct the generalized frame fields E A M satisfying the frame algebrâ (1.4) where£ denotes the generalized Lie derivative in DFT and X AB C are the structure constants of the DD + . Similar to the recent studies on the Poisson-Lie T -duality/T -plurality in the context of DFT [20][21][22], exploiting the relation (1.4), we show that the Jacobi-Lie T -plurality is a symmetry of type II DFT.
At the level of the supergravity (or more precisely, DFT), the proposed Jacobi-Lie Tduality is indeed a symmetry of the equations of motion even if the Ramond-Ramond (R-R) fields or spectator fields are present. However, at the level of the string sigma model, due to the presence of the scale factor, we find difficulty in showing the covariance of the equations of motion under the Jacobi-Lie T -plurality. We discuss this issue from several approaches and also discuss the relation to the Jacobi-Lie T -duality proposed in [17].
This paper is organized as follows. In section 2, after introducing the Leibniz algebra DD + , we explain how to construct the Jacobi-Lie structures and the generalized frame fields from the DD + . We find that the generalized frame fields several concrete examples of the Jacobi-Lie T -plurality with and without the R-R fields or the spectator fields. In section 4, we discuss the issue of the Jacobi-Lie T -plurality in the string sigma model. Section 5 is devoted to conclusion and discussion.

Jacobi-Lie structures
In this section, we propose a Leibniz algebra DD + and construct several quantities, such as the Jacobi-Lie structure, which play an important role in the Jacobi-Lie T -plurality. In section 2.3, we clarify the relation between the DD + and the Jacobi-Lie bialgebra studied in [23][24][25][26].
Several examples are given in section 2.4. In section 2.5, we comment on a relation between DD + and embedding tensors in half-maximal 7D gauged supergravity.

Algebra
A (classical) Drinfel'd double can be defined as a 2D-dimensional Lie algebra d which admits an adjoint-invariant metric ·, · and allows a decomposition d = g ⊕g, where g andg form Lie subalgebras that are maximally isotropic with respect to ·, · . We choose the basis T a ∈ g and T a ∈g such that the metric becomes T a , T b = δ b a , and denote the subalgebras as [T a , T b ] = f ab c T c and [T a , T b ] = f c ab T c . Then, from the adjoint invariance we can determine the mixed-commutator as The adjoint-invariant metric can be expressed as and we raise or lower the indices A, B by using η AB and its inverse η AB . Now, let us introduce the Leibniz algebra DD + , We keep assuming that g andg are maximally-isotropic Lie subalgebras but relax the adjointinvariance as in Eq. (1.3). We then find that the structure constants should have the form where F AB C = F ABD η DC , F ABC = F [ABC] , and F ABC has the only non-vanishing components F ab c and F a bc . Defining f ab c and f c ab through T a • T b = f ab c T c and T a • T b = f c ab T c , we can parameterize F ABC as where Z A = (Z a , Z a ) . By substituting these into Eq. (2.4), we obtain the algebra (1.1).
The closure conditions, or the Leibniz identities, require the following identities for the structure constants: (2.10)

Generalized frame fields
Here we construct the generalized frame fields E A M . We introduce a group element g = e x a Ta and define the left-/right-invariant 1-forms as (2.11) Their inverse matrices are denoted as v m a and e m a (v m a ℓ b m = δ b a = e m a r b m ). We then consider the adjoint-like action as and define It turns out that this matrix M A B can be parameterized as where π ab is an antisymmetric field: π ab = −π ba .
Similar to the case of the Drinfel'd double [27] (see also [9] for a general discussion), we find that a a b , π ab , and ∆ satisfy the algebraic identities 18) and the differential identities where D a ≡ e m a ∂ m . Combining these identities, we also find Here we have defined π mn ≡ e 2∆ π ab e m a e n b , (2.23) which turns out to be a Jacobi-Lie structure.
Now we define the generalized frame fields as and obtain If Z a = 0 , these generalized frame fields satisfy the relation by means of the generalized Lie derivative in DFT, where ∂ M ≡ (∂ m ,∂ m ) are partial derivatives with respect to the doubled coordinates x M ≡ (x m ,x m ) and the indices M, N are raised or lowered with the metric η M N (which is the same matrix as η AB ). In the presence of Z a , we need to modify the generalized frame fields as whereσ is supposed to be positive. If thisσ satisfies 29) we find that the new generalized frame fields satisfy the desired relation (2.26).
Since the modified generalized frame fields depend on the dual coordinatesx m , one may be concerned about the section condition (i.e., a consistency condition in DFT). However, we can easily show that the section condition is not broken. As we discuss later, the supergravity fields are constructed from E A M which is composed of the fields {∆,σ, e m a , π mn } . 2 Using £ Z = Z a £ va , the differential identities, and the Leibniz identities, we find Therefore, Z is a Killing vector field and we can choose the coordinate system such that Z = ∂ w is realized. Then all of the fields φ are independent of the coordinate w . In this 2 In the presence of the dilaton and the Ramond-Ramond fields, there are additional fields which should be chosen such that the section condition is not broken.
coordinate system, we can explicitly findσ = −2w + const., and then the section condition reduces to This is indeed satisfied because fields φ are independent of w due to the Killing equation.
Let us also show several properties of the bi-vector field π ≡ 1 2 π mn ∂ m ∧ ∂ n . By using the differential and algebraic identities, we can show where E ≡ −2 Z a e a and we have defined the Schouten-Nijenhuis bracket for a p-vector v and a q-vector w as 33) or more explicitly, The first property is equivalent to the absence of the non-geometric R-flux 35) and the second one follows from for any functions f and g , the Jacobi identity (2.39) In particular, by choosing a constant function f = const. , the Jacobi identity requires £ E π mn = 0 , and then the Jacobi identity is equivalent to the conditions (2.32). The bracket (2.37) is known as the Jacobi bracket and accordingly, the pair of the bi-vector field π mn and the vector field E satisfying Eq. (2.32) is called the Jacobi structure. In particular, when E = 0 , the Jacobi bracket/structure reduces to the usual Poisson bracket/structure. To consistently define the Jacobi structure on a group manifold G = exp g , properties (2.22), called the multiplicativity [24], need to be satisfied. In our construction, the multiplicativity is automatically satisfied, and then this kind of Jacobi structure is called the Jacobi-Lie structure.
As it has been studied in [23,24,26], the Leibniz identity (2.9) can be regarded as a cocycle condition, and it is automatically satisfied if we consider the coboundary ansatz which are known as the generalized classical Yang-Baxter equations [24]. For this type of algebra, we can find the solution of the differential equation (2.22) as We note that this type of Jacobi-Lie structures (associated with coboundary-type algebras) has been studied in [24] (see also [17,26]).

Jacobi-Lie bialgebra
Let us explain the relation between DD + and the Jacobi-Lie bialgebra studied in [23][24][25][26]. We begin with a Lie algebra g with commutation relation [T a , T b ] = f ab c T c . We introduce the dual space g * spanned by {T a } and suppose that they form a Lie algebra [T a , T b ] = f c ab T c .
We introduce the differentials d and d * which acts on g * and g as and 1-cocycles X 0 ∈ g and φ 0 ∈ g * satisfying d * X 0 = 0 and dφ 0 = 0 . We then define and a bracket [·, ·] φ 0 for x ∈ ∧ p g and y ∈ ∧ q g as where [·, ·] is the algebraic Schouten bracket and ι φ 0 denotes the contraction. Using these, we can define a Jacobi-Lie bialgebra as a pair ((g, φ 0 ), (g * , X 0 )) which satisfies for any elements x, y ∈ g . If we expand X 0 and φ 0 as They are exactly the same as the Leibniz identities of the DD + under the identification This shows that there is a one-to-one correspondence between a Leibniz algebra DD + and a Jacobi-Lie bialgebra. In [25], by using a generalized Courant bracket, commutation relations are introduced, but in general, the Jacobi identities are not satisfied and this bracket does not define a Lie algebra. Rather, this can be regarded as the antisymmetric part of the Leibniz algebra DD + , As we discussed in section 2.2, a DD + allows us to systematically construct the Jacobi-Lie structure π mn for a general Jacobi-Lie bialgebra. In [17], a similar construction has been attempted by using the commutation relations (2.51). However, due to the absence of the symmetric part X (AB) C of the structure constants, it was not successful, and only the coboundary-type algebras have been studied, where π mn has the simple expression (2.42). A DD + also allows us to obtain the scale factor ∆ from a straightforward computation of the matrix M A B , and these are the advantage of our approach based on the Leibniz algebra. In the next subsection, as a demonstration, we explicitly compute the Jacobi-Lie structures for several concrete examples.

Examples of Jacobi-Lie structures
The low-dimensional Jacobi-Lie groups have been classified in [25], and in particular, classifications of the coboundary-type Jacobi-Lie groups have been given in [26]. For the coboundarytype algebras, there is a general formula (2.42) for the Jacobi-Lie structures, and here we consider two examples of Leibniz algebras that are not of the coboundary type.
Using g = e x T 1 e y T 2 e z T 3 , the left-/right-invariant vectors are found as and by computing the matrix M A B , we find (2.55) From∂ mσ = −2 Z a v m a we can easily find σ = 2 αz + const., (2.56) and then we find that the generalized frame fields enjoy the algebra£ Another example is ((III, −2X 1 ), (III.ii, −(X 2 + X 3 ))) of [25], which corresponds to Using g = e x T 1 e y T 2 e z T 3 , the left-/right-invariant vectors are found as (2.58) From the matrix M A B and∂ mσ = −2 Z a v m a , we find π = (z − y) ∂ y ∧ ∂ z , e −2∆ = e 2x ,σ =ỹ +z + const., (2.59) and then the generalized frame fields satisfy the algebra£ In this way, for a given Leibniz algebra, we can easily compute the Jacobi-Lie structure and the generalized frame fields.

Embedding tensor in half-maximal 7D gauged supergravity
As a side remark, we here clarify the relation between six-dimensional DD + s and the embedding tensors in half-maximal 7D gauged supergravity. In [28], embedding tensors in halfmaximal 7D gauged supergravity have been classified, where the duality group is O(3, 3)×R + .
In our convention, their embedding tensor can be expressed as where the non-vanishing components are The possible values of Q ij ,Q ij , and ξ 0 have been classified in Table 2 of [28] and there are 13 inequivalent solutions, which are called orbits (see Appendix A).
Using Eq. (2.60). we can define a Leibniz algebra Due to the presence of F abc and F abc , this is not an algebra of a DD + . However, as we explain in Appendix A, by performing an O(3, 3) redefinition of the generators, most of the 13 orbits can be mapped to some DD + s. As a demonstration, let us take orbit 10, where Q ii andQ ii are given by Performing a redefinition of the generators we find that the structure constants become For example, if ξ 0 = sin α or ξ 0 −sin α cos α = −1 is realized, this is equivalent to a Jacobi-Lie bialgebra ((VI 0 , b X 3 ), (I, 0)) or ((III, b X 1 ), (I, 0)) given in Table 7 of [25], respectively. By choosing another matrix C A B , we may also find another Jacobi-Lie bialgebra classified in [25].
In this sense, the flux algebra given in Eqs. (2.60) and (2.61) can be mapped to a DD + . Then, as we discussed in the previous section, we can systematically construct the generalized frame fields (or twist matrix) by using the Jacobi-Lie structure.
A similar analysis can be carried out for any (half-)maximal d-dimensional supergravities, because the T -duality-covariant flux F ABC is always contained in the embedding tensor and the role of Z A can be played by the trombone gauging [29][30][31]  appear as some components of the quadratic constraints studied in [28,32,33].
As a natural extension, non-Abelian U -duality associated with EDA has been discussed in [4][5][6][7][8][9][10]16], and several examples of the non-Abelian U -duality have been found in [11]. Here, we show that the non-Abelian duality based on a DD + , i.e., the Jacobi-Lie T -plurality, is a symmetry of the DFT equations of motion.

Generalized fluxes
In type II DFT, the bosonic fields in the NS-NS sector are the generalized metric and the DFT dilaton whereĤ AB is constant, ϕ(x) is a certain function, and we have defined When the target space is of this form, this background is called Jacobi-Lie symmetric.
If we parameterize the constant matrixĤ AB aŝ by comparing the parameterization (3.1) with (3.2), the metric and the B-field can be ex- They can be also expressed as The standard dilaton Φ can be found as The structure constants Z A , which are not present in the Poisson-Lie T -duality, produce the overall factor e −2ω both in the metric and the B-field. We find that E mn satisfies Here, let us comment on the difference between our proposal and the one studied in [17].
In [17], the metric and the B-field are identified as for which we have The difference is only in the overall factor e 2ω . Below, we show the covariance of the equations of motion under the Jacobi-Lie T -plurality by adopting the former choice g mn + B mn = E mn and using the dilaton (3.7).
The generalized fluxes associated with E A M are defined as Using the algebraic and the differential identities, we find where F ABC is the one given in (2.6) and we have used Eq. (3.2). When f b ba does not vanish, we can remove the last term by making a replacement [21] and then the single-index flux becomes In the following, we suppose that F A is constant.

Covariance of the equations of motion
In general, the equations of motion of DFT are given by Here, R and G AB , under the section condition, can be expressed as In our setup, we find important relations and we obtain where R and G AB are constants of the form Then the equations of motion simply become R = 0 and G AB = 0 , which are manifestly covariant under the O(D, D) rotation The transformations in the first line are equivalent to a redefinition of generators symmetry is the Jacobi-Lie T -plurality and is a manifest symmetry of DFT.
For later convenience, let us also find the transformation rule of the generalized Ricci and η ab ≡ ηāb ≡ diag(−1, 1, . . . , 1) . We suppose that the double vielbein is transformed as under the Jacobi-Lie T -duality, and then the transformation rule for is found as We find that the only non-vanishing components of G AB are G ab , and using these, we can express the generalized Ricci tensor as Then, using (3.28), we find the transformation rule of the generalized Ricci tensor S M N as Namely, under the Jacobi-Lie T -plurality, or a local O(D, D) rotation of the generalized metric, the generalized Ricci tensor transforms as Unlike the case of the Poisson-Lie T -duality, the generalized Ricci tensor is transformed by a local O(D, D) × R + rotation. As we discuss later, this additional R + transformation makes the transformation rule of the R-R fields slightly non-trivial.

Comments on a subtle issue
Here, we comment on an issue that may arise in the presence of Z a and f b ba .
Firstly, we consider the case where two vectors I ≡ 1 2 f b ba v a and Z = Z a v a are proportional to each other (which includes the case where I = 0 or Z = 0). Since Z is a Killing vector field, we can choose a coordinate system such that Z = c Z ∂ w and I = c I ∂ w (where c Z and c I are constants). In such a coordinate system, recalling∂ mσ = −2 Z m , we find where c 0 is a constant. Now we consider the following three cases.
1. c I = 0 and c Z = 0 In this case, the shift (3.13) is not necessary, and we can chooseσ = 1 by a redefinition of ϕ(x) . Then the metric and the B-field are independent of the dual coordinates.
By recalling Eq. (3.14), they are equivalent to In particular, if ϕ is independent of the dual coordinates, the DFT solution corresponds to a solution of the usual supergravity.
2. c I = 0 and c Z = 0 Again we can chooseσ = 1 , and then the metric and the B-field are independent of the dual coordinates. The shift (3.13) corresponds to introducing the dual-coordinate dependence into the dilaton [34,35] d → d + c Iw . Namely, the dilaton becomes The section condition is satisfied if £ I e −2ϕ(x) |det ℓ a m |) = 0 and Eq. (3.34) are satisfied. In this case, for example if ϕ is independent of the dual coordinates, the DFT solution corresponds to a solution of the generalized supergravity equations of motion [35][36][37].
Then, the dilaton becomes Here, the Leibniz identities ensures £ Z |det ℓ a m | = 0 , and the section condition is satisfied if £ Z ϕ = 0 (⇔ Z a F a = 0 = π ab Z a F b ) and Eq. (3.34) are satisfied. Even if this is a solution of DFT, this does not correspond to a solution of the usual (or the generalized) supergravity because the metric and the B-field depend on the dual coordinates.
If F A = 0 (or ϕ = 0), we do not need to care about the section condition: only the second case requires a non-trivial relation In such a case, we cannot find the DFT dilaton, and it does not correspond to a solution of the usual DFT.
Secondly, let us consider the problematic case where two vector fields I and Z are linearly dependent. Using the Leibniz identities (2.8)-(2.10), we can show that they commute with each other [I, Z] = 0 . Since Z is a Killing vector field, if we suppose that I is also a Killing vector field, we can choose a coordinate system such that Z = ∂ w and I = ∂ z . In such a coordinate system, we findσ = c 0 − 2w (c 0 : arbitrary constant) and we also find 1 After the sift (3.13), the derivative of the dilaton becomes where F M is defined in (3.14).  In the Poisson-Lie T -plurality, there is a prescription to find ϕ ′ (x ′ ), which is based on a coordinate transformation on the Drinfel'd double [3]. However, unlike a Lie algebra which can be exponentiated to a Lie group, it is not clear how to globally extend the Leibniz algebra DD + to a group-like space. Then the procedure of [3] does not work and the function ϕ ′ needs to be found by solving the differential equation

An example without Ramond-Ramond flux
Let us consider an eight-dimensional Leibniz algebra with which is a direct sum of the six-dimensional Leibniz algebra ((IV, −4X 1 ), (I, 0)) of [25] and a two-dimensional Abelian algebra. Using a parameterization g = e x T 1 e y T 2 e z T 3 e w T 4 , we find (3.40) Computing the matrix M A B , we find Then, using the constant matriceŝ we obtain a 4D metric ds 2 = 2 e 3x dx (dz + x dy) + e 2x dy 2 + e 4x dw 2 . (3.43) In order to find a solution of DFT, we choose the function ϕ as which yields Then the DFT dilaton and the standard dilaton become We can check that this dilaton and the metric (3.43) satisfy the equations of motion. In the following, we consider the Jacobi-Lie T -pluralities of this solution.

Another Jacobi-Lie T -plurality
Here we consider another O(4, 4) transformation We then obtain the algebra with The six-dimensional part of this algebra is known as ((VI 2 , −4X 1 ), (II, 0)) . Using the parameterization, g = e x T 1 e y T 2 e z T 3 e w T 4 , we obtain (3.53) We can compute several quantities as The associated supergravity fields are found as and this is a solution of the supergravity.

Jacobi-Lie T -duality
To provide an example with Z a = 0 , let us consider the T -dual of the previous example. The non-vanishing structure constants are (3.57) Since we find 1 2 f b ba = −Z a , this example corresponds to the third case discussed around Eq. (3.37). By using a parameterization g = e x T 1 e y T 2 e z T 3 e w T 4 , we find The flux F A shows that ϕ(x) = − 1 3 lnσ , and by using the formula (3.37), the DFT dilaton is found as This dilaton together with the generalized metric (3.60) satisfies the DFT equations of motion.
Since the metric and the B-field have the dual-coordinate dependence through the overall factorσ , this is not a solution of the usual (or the generalized) supergravity. However, the section condition is not broken and can be mapped to a DFT solution that does not depend on dual coordinates.

Another example without Ramond-Ramond flux
To provide a problematic example, let us consider which corresponds to the T -dual of ((I, 0), (III, −(X 2 −X 3 ))). Using a parameterization,

A problematic example
Now we perform the Jacobi-Lie T -duality, T a ↔ T a . The resulting DD + has the structure constants In this case, 1 2 f b ba and Z a are linearly independent, which is problematic as we have discussed.
Using a parameterization g = e x T 1 e y T 2 e z T 3 , we can straightforwardly compute the generalized metric as whereσ ≡ c 0 +ỹ −z . One can check that this satisfies the section condition, ∂ P ∂ P H M N = 0 and ∂ R H M N ∂ R H P Q = 0 , and there is no problem at this stage.
The problem is related to the dilaton. By using ∆(x) = 0 ,σ = c 0 +ỹ −z , and |det ℓ a m | = 1, the general formula (3.2) gives By considering the shift (3.13) and using v m a = δ m a , the derivative of the DFT dilaton becomes We can easily compute 1 into the equations of motion, we find that the DFT equations of motion are indeed satisfied.
A problem is that the section condition is broken by the DFT dilaton, Another problem is that we cannot find the DFT dilaton d that solves the differential equation (3.72). Consequently, this configuration cannot be regarded as a solution of DFT.

Ramond-Ramond fields
We here introduce the R-R fields by considering the case D = 10 . In the presence of the R-R fields, the equations of motion for the generalized metric and the DFT dilaton become where E M N denotes the energy-momentum tensor of the R-R fields. Obviously, if we transform the energy-momentum tensor as the equations of motion for the generalized metric transform covariantly as where U ≡ (E M A ) and S U is a matrix representation of U in the spinor representation (see [22] for our convention). The presence of e ω is the only difference from the Poisson-Lie T -duality.
The O(10, 10) spinor |F is constant, and in type IIA/IIB theory, it can be expanded as where |0 is the Clifford vacuum satisfying Γ a |0 = 0 . Under the ansatz (3.77), the equations of motion of the R-R fields become the algebraic relation For convenience, let us also express (3.77) in terms of the differential form. By using a polyform F ≡ p : even/odd in type IIA/IIB theory, we have |det a a b | 1 2 e 1 2 π ab ιaι b p : even/odd 1 p!F a 1 ···ap r a 1 ∧ · · · ∧ r ap . (3.82) Note that here we are using the field strength in the A-basis (which satisfies dF = 0) and this is related to the one in C-basis as that satisfies the standard Bianchi identity when G is independent of the dual coordinatesx m .

An example with Ramond-Ramond fluxes
Let us consider a 20-dimensional DD + with the structure constants The non-trivial subalgebra generated by {T 1 , T 2 , T 3 } are known as ((III, −4X 1 ), (I, 0)) . Using the parameterization g = e x T 1 e y T 2 e z T 3 e w 4 T 4 · · · e w 10 T 10 , the non-trivial part of v m a and e m a are found as (the other components are just v a = e a = ∂ a ) (3.86) We introduce constantŝ and then, by using ∆ = −2x and supposing ϕ = 0, the supergravity fields are found as ds 2 = e 4x dx 2 + dx (dy − dz) + ds 2 T 7 + e 2x dx (dy + dz) + (dy + dz) 2 , where ds 2 T 7 ≡ dw 2 4 + · · · + dw 2 10 is a seven-dimensional flat metric. This is a solution of type IIB * supergravity. Now we consider a generalized Yang-Baxter deformation with (3.89) The resulting DD + has the structure constants The structure constants f 1 23 produces the Jacobi-Lie structure π = η 2 (1 − e −2x ) ∂ y ∧ ∂ u and the supergravity fields are

Jacobi-Lie T -plurality with spectator fields
The inclusion of the spectator fields is straightforward similar to the case of the Poisson-Lie T -duality/T -plurality (see Appendix B of [22]). Here, instead of repeating the presentation of [22], we only comment on some non-trivialities that are specific to the Jacobi-Lie T -plurality.
We consider a ten-dimensional spacetime with the "internal coordinates" x m (m = 1, . . . , D) and the "external coordinates" y µ (µ = D + 1, . . . , 10). In the string sigma model, the scalar fields y µ (σ) are called the spectator fields because they are invariant under the non-Abelian duality. We formally double all of the directions, and the generalized coordinates are given by in the presence of the spectator fields, but without the R-R fields, are given by The difference is that V A A (y) is no longer constant and that the dilaton also acquires the y-dependenced(y) . By following the same discussion as [22], we can show that the O (D, D) transformation which rotates the internal indices is a symmetry of the equation of motion.
When the R-R fields are also present, the symmetry becomes slightly subtle. In the presence of the spectator fields, the tensor G AB becomes M ∂ M and the fluxes contain both the external and internal parts: The existence of the two options are specific to the Jacobi-Lie T -plurality, and these two are degenerate in the case of the Poisson-Lie T -duality (where ω = 0). In the next subsection, we present an example using the latter option (3.98).

An example with spectator fields
We consider an eight-dimensional DD + (D = 4) with the structure constants given in Eq. (3.85).
We introduce the ten-dimensional coordinates and y µ are the spectator fields. Using the parameterization g = e x T 1 e y T 2 e u T 3 e v T 4 , we obtain the left-/right-invariant vector fields as given in Eq. (3.86). We choose the metricĝ ab (y) , dilatond(y) , the R-R field |F (y) , and ϕ(x) aŝ ,β ab = 0 , e −2d = cos r cos ξ sin 3 r sin ξ z 5 , where the metric g S 5 on S 5 corresponds to the line element ds 2 S 5 ≡ dr 2 + sin 2 r dξ 2 + cos 2 ξ dφ 2 1 + sin 2 ξ dφ 2 2 + cos 2 r dφ 2 3 . (3.102) Using π ab = 0 and ∆ = −2 x, the generalized frame fields become By acting the twist, we find that this is the AdS 5 ×S 5 solution of type IIB supergravity, Here we have used e −ϕ(x) e D−1 = 4), and The deformed geometry is This also satisfies the type IIB supergravity equations of motion.
In order to perform more interesting Jacobi-Lie T -plurality, the classification of the sixdimensional DD + will be useful. The classification of the Jacobi-Lie bialgebra has been done in [25] but which bialgebras are in the same orbit O(D, D) rotations have not been studied.
If such a classification is worked out, we may find more dual geometries from the AdS 5 ×S 5 solution (3.104).

Jacobi-Lie T -plurality in string theory
In the string sigma model, we can clearly see the symmetry of the Poisson-Lie T -duality by using a formulation called the E-model [38]. The E-model is defined by a Hamiltonian and the current algebra where E A M are the generalized frame fields satisfying£ E A E B = −F AB C E C with F AB C the structure constants of a Drinfel'd double. Here, the currents have been identified as where p m are the canonical momenta associated with x m . The current algebra (4.2) is simply a rewriting of the canonical commutation relation we get a new generalized frame fields E ′ M satisfies the canonical commutation relation (4.6). The Hamiltonian for string theory on the dual geometry can be expressed as whereĤ ′AB ≡ (C −1 ) C A (C −1 ) D BĤCD and the dual currents satisfies the algebra Then the currents j ′ A follow the same time evolution as C A B j B , and we can clearly see the covariance of the string equations of motion. In [27], the currents j A was regarded as the phase-space variables and the Poisson-Lie T -duality/T -plurality j A → j ′ A that preserves the Hamiltonian was regarded as a canonical transformation. Now let us consider the case of the Jacobi-Lie T -plurality. Again the generalized metric is expressed as and F AB C is the one given in (2.5). Then introducing the currents we obtain the Hamiltonian and the current algebra as Due to the appearance of the explicit x-dependence in e ω(x(σ)) , we cannot treat the currents as the phase-space variables, but as complicated functions of x(σ) and their canonical conjugate momenta. Then the Hamiltonian also needs to be regarded as a non-linear function.
Consequently, the covariance under the Jacobi-Lie T -plurality is not manifest.
Let us also discuss the covariance from another perspective. If we start with the action the equations of motion can be expressed as where J a ≡ v m a g mn * dx n + B mn dx n . If we identify the metric and the B-field as g mn + B mn = E mn , by using Eq. (3.10), the equations of motion can be rewritten in a suggestive form [17] However we cannot say anything more from this relation.
In the case of the Poisson-Lie T -duality, where ∆ = 0 and Z a = 0 , we can regard the relation (4.17) as a Maurer-Cartan equation and identify the current J a as the right-invariant 1-form In the case of the Jacobi-Lie T -duality of [17], due to the presence of ∆ in Eq.
In this case, there is no scale factor, but due to the presence of the last term, this again cannot be regarded as a Maurer-Cartan equation. According to the above considerations, we suspect that the Jacobi-Lie T -plurality is not a symmetry of the string sigma model.
One of the reasons for the issue may be that the DD + is a Leibniz algebra instead of a

Conclusions
In this paper, we proposed a Leibniz algebra DD + and showed that this provides an alternative description of the Jacobi-Lie bialgebra. Extending the standard procedure developed in the Poisson-Lie T -duality, we showed that a DD + systematically constructs a Jacobi-Lie structures and the generalized frame fields satisfying£  [40][41][42].
In M-theory, the exceptional Drinfel'd algebra (associated with the SL(5) duality group) has been found as If we decompose the index as a = {ȧ, ♯} and assume fȧ˙b ♯ = 0 , we find that the generators {Tȧ, Tȧ ≡ Tȧ ♯ } satisfy the subalgebra This is noting but the DD + under the identifications, fȧ˙bċ = −fȧ˙bċ ♯ , Zȧ = 0, and 2 Zȧ = 3 Zȧ − fȧ ♯ ♯ . Similarly, the extended Drinfel'd algebra in the type IIB picture also contains the DD + as a subalgebra. Thus, the Jacobi-Lie T -plurality is a subset of the proposed Nambu-Lie U -duality. 6 An issue in the Nambu-Lie U -duality is that the equations of motion of the exceptional field theory are complicated and the covariance under the Nambu-Lie U -duality cannot be easily proven. The results of this paper show that the non-Abelian duality works as a solution generating transformation even when the Z A is present. Further steps towards the proof of Nambu-Lie U -duality will be taken in future work.
Another future direction is to study a U -duality extension of the Jacobi-Lie structure.
For this purpose, we need to study the Nambu-Jacobi structure [45] on a group manifold.
In this paper, we have constructed the Jacobi-Lie structure π and E from a DD + , and the vector field E ∝ Z a e a is associated with the vector Z A = (Z a , Z a ) . In the case of the EDA (in the M-theory picture), π is replaced by a tri-vector π (3) and E will be replaced by a bivector E (2) ∝ Z ab e a e b because Z A is replaced by Z A = (Z a , Z a 1 a 2 √ 2! ). In the literature, the non-Abelian U -duality is studied by assuming Z a 1 a 2 = 0 , but this assumption may not be necessary. It will be an interesting future work to keep Z a 1 a 2 to find a generalized non-Abelian U -duality. It is also interesting to study the associated generalized Yang-Baxter deformation.
A Embedding tensors in half-maximal 7D SUGRA and DD + In this appendix, we conduct a detailed study of the relationship between the embedding tensors in half-maximal 7D gauged supergravity and the DD + . Among the 13 inequivalent orbits classified in [28], we show that orbits 2, 3, 5, 7, . . . , 13 can be mapped to some DD + s by performing O(3, 3) redefinitions of generators T A → C A B T B . For each orbit, the matrix C A B (which is not unique) is found by trial and error. For orbit 4 or 6, only when α = 0, we find such a matrix C A B but failed to find such matrix for α = 0. For orbit 1, as we explain below, we conclude that this is not related to any DD + .
In the following, we use a short-hand notation, In addition, as was classified in [43,44], there are 22 six-dimensional Drinfel'd doubles, which are called DD1, . . . , DD22, and we use the notation in the following. As we show in the following, all of these are related to some embedding tensors with ξ 0 = 0 (recall that a DD + reduces to a Drinfel'd double when ξ 0 = 0). The 6D Lie algebra [T A , T B ] = X AB C T C has been identified as SO(4) for α = π 4 or SO(3) (times three-dimensional Abelian algebra) for α = π 4 . According to the classification of sixdimensional Drinfel'd double [43], there is no Drinfel'd double whose Lie algebra is SO(4) or SO(3) , and thus orbit 1 is not related to any Drinfel'd double. According to [43], this corresponds to a Manin triple (7 0 |5.ii|b) with b = 1 , which corresponds to the Drinfel'd double we get a Drinfel'd double with This corresponds to a Manin triple (7 a |7 1/a |b) with a ≡ 1 t and b = c s = a 1+a 2 . This corresponds to the Drinfel'd double DD3: 3. α < 0 This case is related to the previous case through T 1 → −T 1 and T 2 ↔ T 3 .
As one can see from this example, each orbit of [28] we get a Drinfel'd double with This corresponds to a Manin triple (6 0 |5.iii|b) with b = 1 , which is contained in DD2: The Lie algebra of this Drinfel'd double is isomorphic to SO(2, 2) .
Performing an O(3, 3) transformation with we get a Drinfel'd double with whose Lie algebra is isomorphic to SO(2, 1) .

Orbit 4
Orbit 4 this is mapped to a flux configuration Therefore, the 6D Lie algebra is isomorphic to ISO(3) for any value of α as discussed in [28]. However, since the matrix (A.22) is not an element of O(3, 3) , the redefined generators T ′ A do not have the canonical bilinear form: T ′ A , T ′ B = η AB . According to [43], the only Drinfel'd double whose Lie algebra is isomorphic to ISO(3) is DD5. We have tried to find an O(3, 3) transformation which maps the algebra (A.20) to the Lie algebra of (9|1) but we could not find such an O (3,3) . We thus conclude that orbit 4 is related to a Drinfel'd double only when α = 0 .

Orbit 5
Orbit 5 contains the non-vanishing fluxes Here we consider two cases.

Orbit 6
Orbit 6 contains the non-vanishing fluxes This and the subsequent orbit contain non-vanishing Z A and the structure constants X AB C have the symmetric part: X (AB) C = 0 . If α = 0 , the flux configuration coincides with that of orbit 8 with α = 0 , which can be mapped to a DD + (which reduces to DD15 when ξ 0 = 0).
When α = 0 , we fail to find an O(3, 3) transformation which maps this fluxes into any DD + . 7 To be a little more specific, let us consider the case ξ 0 = 0 . By considering the eigenvalues of the Killing form, the only possible Drinfel'd doubles that may be related to orbit 6 are DD14-DD17. However, we could not find any O(3, 3) transformation which maps the flux configuration (A.30) with ξ 0 = 0 to any of these Drinfel'd double. 8 As we see below, DD14-DD17 rather correspond to orbit 7 or 8. We thus conclude that orbit 6 with ξ 0 = 0 can be related to a Drinfel'd double only when α = 0 . If we consider ξ 0 = 0 , the DD + reduces to a Drinfel'd double, which can be classified as follows depending on the value of α.
2. − π 4 < α < 0 In this case, the algebra can be mapped to the previous one through T 1 → −T 1 and T 2 ↔ T 3 .

(A.45)
When α is negative, we can consider a redefinition, such as T 1 → −T 1 and T 2 → −T 2 , which flips the sign of t , and the Drinfel'd double is always DD9.
This case is the same as the previous case by choosing α = π 4 . When ξ 0 = 0 , the Drinfel'd double has another name If we consider the case ξ 0 = 0 , we obtain the following Drinfel'd doubles.
1. α = 0 Again, the embedding tensor is the same as that of orbit 11 with α = 0.

Summary
We can summarize the result as in Table A [43] can be reproduced from the orbits classified in [28].