Generalizing the $\mathfrak{bms}_{3}$ and 2D-conformal algebras by expanding the Virasoro algebra

By means of the Lie algebra expansion method, the centrally extended conformal algebra in two dimensions and the $\mathfrak{bms}_{3}$ algebra are obtained from the Virasoro algebra. We extend this result to construct new families of expanded Virasoro algebras that turn out to be infinite dimensional lifts of the so-called $\mathfrak{B}_{k}$, $\mathfrak{C}_{k}$ and $\mathfrak{D}_{k}$ algebras recently introduced in the literature in the context of (super)gravity. We also show how some of these new infinite-dimensional symmetries can be obtained from expanded Ka\v{c}-Moody algebras using modified Sugawara constructions. Applications in the context of three-dimensional gravity are briefly discussed.


Introduction
in turn follows from an IW contraction of an sl(2) Kač-Moody algebra. Along the same lines, a Hamiltonian reduction of the flat WZW model leads to the flat Liouville theory as the classical two-dimensional dual for asymptotically flat 3D Einstein gravity, which is BMS 3 invariant [22][23][24]. On the other hand, there is an equivalence between symmetries of ultra-relativistic theory and bms 3 which has been relevant in the extension of the AdS/CFT correspondence [25][26][27][28]. In the last years, generalizations of the conformal and bms 3 algebras together with their Kač-Moody cousins have appeared in the literature in the context of three-dimensional supergravity and higher spin gravity [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45]. Analogously to the pure gravity cases, these extensions turn out to be connected by IW contractions.
A particular characteristic of the IW contraction is that the starting and resulting algebras have the same number of generators. A natural way to generalize the IW contraction in order to obtain algebras of greater dimension than the starting one is given by the Lie algebra expansion method [46][47][48][49][50]. In particular, the S-expansion procedure formulated in ref. [50] combines the structure constants of a given Lie algebra with the inner product of an abelian semigroup and has given rise to a number of interesting new symmetries that can be used to formulate gravity theories coupled to matter [51][52][53][54][55][56][57][58][59][60]. Such symmetries can be classified into three families of algebras called B k , C k and D k . B k algebras have been used to obtain General Relativiy from Chern-Simons and Born-Infeld gravity theories in diverse dimensions [51,53,55,56]. In particular, the B 3 and B 4 algebras correspond to the Poincaré and Maxwell algebras [61][62][63][64][65][66]. It is important to note that B k symmetries can be obtained as IW contraction of the C k algebras [54]. The C k family allows one to relate diverse (pure) Lovelock gravities to Chern-Simons and Born-Infeld gravities [58,60]. Alternatively, B k algebras can be obtained as an IW contraction of another set of algebras called D k symmetries, which correspond to direct sums of the form AdS ⊕ B k−2 [57,59]. Supersymmetric extensions of some of these expanded algebras have been worked out in refs. [67][68][69][70][71][72][73][74], which can also be obtained through the Sexpansion mechanism. It is therefore interesting to study what kind of infinite dimensional symmetries can be obtained as S-expansions of known infinite dimensional algebras. In this paper we put forward such study and present new families of infinite dimensional algebras that can be obtained by applying the semigroup expansion mechanism to the Virasoro algebra. We first show that the centrally extended 2D-conformal algebra and the bms 3 algebra can be obtained as a semigroup expansion of the Virasoro algebra. Then, by using more general semigroups, we construct new families expanded Virasoro algebras that we name generalized 2D-conformal algebras and generalized bms 3 algebras. We also show how these new infinite dimensional symmetries can be related by IW contractions. Interestingly these symmetries correspond to infinite dimensional enhancements of the B k and C k algebras. Additionally, we provide with an infinite dimensional lift of the so called. D k algebras. Finally we study the Sugawara construction connecting expanded Kač-Moody algebras with our expanded Virasoro algebras and present some explicit examples.
The paper is organized as follows: In Sect. 2 we present the general setup to S-expand the Virasoro algebra and obtain the centrally extended 2D-conformal algebra as well as the bms 3 algebra particular cases. In Sect. 3 we show how the expansion procedure can be used to construct a deformed bms 3 algebra which corresponds to an infinite dimensional lift of the Maxwell algebra. In the same way, an infinite dimensional enhancement of the AdS-Lorentz algebra is constructed, which is given by three copies of the Virasoro algebra and can be related to the deformed bms 3 symmetry by an IW contraction. In Sect. 4 we introduce the generalized 2D-conformal algebras and generalized bms 3 algebras. Sect. 5 is devoted to the construction of the infinite dimensional lifts of the D k algebras. In Sect. 6 we present (modified) Sugawara construction that allows one to obtain expanded Virasoro algebras from expanded Kac-Moody algebras in the simplest cases. Finally, future applications of these results in the context of 3D gravity theories and WZW models are discussed in Sect. 7.
2 Centrally extended 2D-conformal algebra and bms 3 algebra as S-expansions The S-expansion method [50] consists in combining the structure constants of a Lie algebra g with the elements of a semigroup S to obtain a new Lie algebra G = S × g. In this section we show that the centrally extended 2D-conformal algebra and the bms 3 algebra can be obtained explicitly as an S-expansion of the Virasoro algebra for suitable semigroups. For details regarding the notation and the S-expansion procedure we refer the reader to the original references [75][76][77][78][79][80][81][82].

Expanding the Virasoro algebra
The starting point of this construction is the Virasoro algebra vir, together with a semigroup S = {λ α }, whose inner product is defined by a 2-selector K γ αβ = K γ βα such that We define an S-expanded Virasoro algebra as the direct product vir h = S × vir, where h = S × sl(2) is the expansion of the sl(2, R) subalgebra of (2.1) generated by subset {ℓ −1 , ℓ 0 , ℓ 1 } 1 . The generators of vir h are given by and satisfy the commutation relations where c αβ denotes a set of central charges given by Note that the finite subalgebra h of vir h is spanned by the subset ℓ (−1,α) , ℓ (0,α) , ℓ (1,α) .

Centrally extended 2D-conformal algebra
The centrally extended conformal algebra in two-dimensions is given by the direct sum of a pair of Virasoro algebras vir ⊕ vir, which we will simply denote as vir 2 , This algebra can be obtained as a particular S-expansion of vir. In fact, let us consider the (semi)group Z 2 = {λ 0 , λ 1 }, whose multiplication law is given by and from which the non-vanishing 2-selectors (2.2) can be read off to be K 0 00 = K 0 11 = K 1 01 = K 1 10 = 1. Denoting the generators (2.3) and the central charges (2.5) of the corresponding expanded algebra as It is easy to see that (2.9) is isomorphic to vir 2 by making the following change of basis which leads to (2.6) with central charges c = 1 2 (c 2 + c 1 ) andc = 1 2 (c 2 − c 1 ).

bms 3 algebra
Consider now the expansion of the Viraroso algebra (2.1) using the semigroup S (1) E = {λ 0 , λ 1 , λ 2 }, whose elements satisfy the multiplication law and where λ 2 ≡ 0 S is the zero element of the semigroup such that 0 S λ α = 0 S . The 0 Sreduced S (1) E -expanded algebra is obtained imposing 0 S ×ℓ (m,α) = 0. Defining the non-vanishing expanded generators (2.3) in the same way as in (2.8), we get which corresponds to the bms 3 algebra [18]. Let us also recall that the bms 3 algebra can be obtained from two copies of the Virasoro algebra as an IW contraction. Writing vir 2 in the form (2.9) and rescaling its generators as leads to (2.12) in the limit σ → ∞. A similar approach is considered in [83] where they implemented the IW contraction using a Grassman parameter. As we will see in the following, this kind of limit procedure will be useful to establish different links between more general expanded Virasoro algebras.

Deformed bms algebra
The centrally extended conformal algebra and its flat limit, the bms 3 algebra, are not the only symmetries that can be obtained using the expansion method. In the present section we present new infinite-dimensional symmetries which are directly obtained as an S-expansion of the Virasoro algebra. In particular, a deformed bms 3 algebra as well as three copies of the Virasoro algebra (vir 3 ) can be obtained, where the former corresponds to an IW contraction of the latter.

Deformed bms 3 as an S-expansion
Let us consider the semigroup S (2) E = {λ 0 , λ 1 , λ 2 , λ 3 }, whose elements satisfy Interestingly, the Maxwell algebra in (2+1) dimensions is spanned by the generators J 0 , J 1 , J −1 , P 0 , P 1 , P −1 and Z 0 , Z 1 , Z −1 . This can be made explicit in terms of generators {J a , P a , Z a } obtained through the following change of basis 2 This means that the deformed bms 3 algebra (3.3) corresponds to an infinite-dimensional lift of the (2 + 1)-dimensional Maxwell algebra in the very same way as the algebras vir 2 and bms 3 are infinite-dimensional lifts of the AdS and the Poincaré algebras in 2 + 1 dimensions respectively.
3.2 Deformed bms 3 algebra as a limit of vir 3 Let us consider now S M = {λ 0 , λ 1 , λ 2 } as the relevant abelian semigroup, whose elements satisfy the following multiplication law E semigroup, there is no zero element in this case. Adopting the same notation (3.2) for the generators of the S M -expanded algebra, we find the following commutation relations

6)
2 In this case the Maxwell algebra is realized with a non-diagonal Minkowski metric η ab = Note that the AdS-Lorentz algebra in 2+1 dimensions, also known as the semi-simple extension of the Poincaré algebra [84], is the subalgebra of (3.6) spanned by the generators J 0 , J 1 , J −1 , P 0 , P 1 , P −1 and Z 0 , Z 1 , Z −1 . This can be explicitly seen using the change of basis (3.4), showing that (3.6) defines and infinite dimensional lift of the AdS-Lorentz algebra in 2 + 1 dimensions.
Remarkably, there is a redefinition of the generators of (3.6) that allows to see its true algebraic structure. In fact, considering the change of basis three copies of the Virasoro algebra, which will be denoted as vir 3 , are revealed where the central extensions are given by Additionally, as in the case of the bms 3 and the 2D-conformal algebra, there is a limit procedure relating vir 3 and the deformed bms 3 algebra through an IW contraction. In fact, the following rescaling of the generators of (3.6), leads to the deformed bms 3 algebra (3.3) in the limit σ → ∞.

Generalized expanded Virasoro algebras
In the previous sections we have seen how the S expansion mechanism allows one to obtain the centrally extended 2D-conformal algebra and the bms 3 algebra from the Virasoro algebra. In the context of three-dimensional gravity, the centrally extended 2D-conformal algebra and the bms 3 algebra correspond to infinite dimensional lifts of AdS and the Poincaré symmetries in 2 + 1 dimensions. Generalizing this results we have subsequently shown how to construct infinite dimensional lifts of the Maxwell and the AdS-Lorentz algebras in 2 + 1 dimensions, which correspond a deformed bms 3 symmetry in the former case and to three copies of the Virasoro algebra in the latter. As it has recently been pointed out in Refs. [54,55,57,68], the Poincaré and the AdS algebras as well as the Maxwell and the AdS-Lorentz algebras correspond to particular cases of the B k and C k algebras for k = 3 and k = 4 respectively. Such families of algebras have been of particular interest in the context of gravity. Indeed, as was shown in Refs. [51,53,56], General Relativity can be obtained as a particular limit of a Chern-Simons and a Born-Infeld gravity theory using the B k symmetries. On the other hand, the C k algebras allow one to recover the pure Lovelock Lagrangian from Chern-Simons and Born-Infeld theories [58,60].
The results obtained up to this point clearly suggest that, in the same way as their respective finite subalgebras, the bms 3 and vir 2 algebras as well as the deformed bms 3 and the vir 3 algebras should correspond to particular cases of certain families of generalized infinite-dimensional symmetries. In this section we present the general scheme that leads to such families of expanded Virasoro algebras.

Generalized bms 3 algebras
. . , λ k−1 } be the finite abelian semigroup whose elements satisfy the following multiplication law where we have defined c α+β+1 ≡ c αβ . Following the notation introduced in section 2, the algebra (4.2) will be denoted by vir B k , as the subalgebra h generated by ℓ (−1,α) , ℓ (0,α) , ℓ (1,α) corresponds to the the B k algebra in 2 + 1 dimensions [51,85]. It is easy to see that (4.1) always contains an abelian ideal spanned by the subset of generators For this reason the vir B k algebra will be referred to as generalized bms 3 algebra. This algebra corresponds to an infinite dimensional lift of the B k algebra in 2 + 1 dimensions, which can be made explicit by redefining the generators in the form where i takes even values andī takes odd values. Here we identify the following cases: • For k − 2 = 2N the abelian ideal A is generated by Using the definition (4.4), one to write (4.2) in the form As mentioned before, the B k algebra in 2 + 1 dimensions is a subalgebra of (4.5) spanned by the generators Additionally, when written in this form it is straightforward to see that setting k = 3 leads to the bms 3 algebra (2.12), while k = 4 reproduces the deformed bms 3 algebra (3.3). Thus, bms 3 and its corresponding deformation can be classified into the infinite family of generalized bms 3 algebras vir B k .

Generalized 2D-conformal algebras
Another family of expanded Virasoro algebras can be obtained by choosing a different semigroup. Let us consider S (k−2) M = {λ 0 , λ 1 , . . . , λ k−2 } as the relevant abelian semigroup whose elements satisfy Then the S (4.7) and corresponds to vir C k , as its subalgebra h is given by the C k algebra in 2 + 1 dimensions [58,60]. This algebra corresponds to an infinite dimensional lift of the C k algebra, which can be explicitly seen by redefining the generators in the form (4.4), yielding where {· · · } means As remarked before, the C k algebra in 2+1 dimensions is the subalgebra of vir C k spanned by the generators J i 0 , J i 1 , J i −1 and Pī 0 , Pī 1 , Pī −1 . When written in the form (4.8) it is clear that setting k = 3 leads to the centrally extended 2D-conformal algebra (2.9), while the case k = 4 leads to the vir 3 algebra (3.6). Therefore vir C k will be referred to as (centrally extended) generalized 2D-conformal algebra. As in the cases k = 3 and k = 4 studied in the previous sections, the generalized 2D-conformal algebra can be related to the generalized bms 3 one through an IW contraction. In fact, rescaling the generators of (4.8) in the form leads to the generalized bms 3 algebra (4.5) in the limit σ → ∞.
The fact that the vir C k reduces to two and three copies of the Virasoro algebra in the cases k = 3 and k = 4, respectively, might make one think that it could generally be written as k − 1 copies of the Virasoro algebra. However this is not true. Let us consider, for instance, the vir C 5 algebra. Renaming its generators as J 0 m ≡ J m , P 1 m ≡ P m , J 2 m ≡ Z m and P 3 m ≡ R m , this algebra can be directly read off from (4.8) to be which cannot be redefined as four copies of the Virasoro algebra by means of a generalization of (2.10) or (3.7).

Infinite dimensional D k -like algebras
In [57] new expanded algebras were presented as a family of Maxwell-like algebras. Inspired by this construction, in this section we consider the expansion of the Virasoro algebra by means of the semigroup S (k−2) D , defined by the product rule Using the notation (4.4) for the expanded generators, the S where {· · · } means These algebra corresponds to vir D k , as their subalgebra h is given by the D k algebra in 2 + 1 dimensions [57] and provides with an infinite dimensional lift of it. Interestingly, this kind of algebras can be written as the direct sum of two copies of the Virasoro algebra and a generalized bms 3 algebra, i.e, vir D k = vir 2 ⊕ vir B k−2 , when an appropriate change of basis is considered. Furthermore, an IW contraction of vir D k using the rescaling (4.10) leads to the generalized bms 3 algebra vir B k . In the following, a few simple examples will be worked out.

vir 2 ⊕ bms 3
The simplest case to consider 3 is k = 5, for which (5.1) yields the vir D 5 algebra: where we have defined The semigroup (5.1) is defined for k > 3 and k = 4 just gives the semigroup S M , which was already studied in section 3.
The Maxwell-like algebra D 5 in 2 + 1 dimensions [57] is spanned by the generators J 0 , J 1 , J −1 , P 0 , P 1 , P −1 , Z 0 , Z 1 , Z −1 and R 0 , R 1 , R −1 . The algebraic structure of the vir D 5 algebra can be made manifest by performing a suitable change of basis. Indeed, the following redefinition reproduces the centrally extended 2D-conformal algebra (2.6) with central charges leads to the bms 3 algebra (2.12) with central charges c 1 = 1 2 (c 1 − c 3 ) and c 2 = 1 2 (c 2 − c 4 ). Since the set of generators L m ,L m commutes with the set J m ,P m , this shows that the vir D 5 algebra is given by the direct sum of these two subalgebras, namely, vir 2 ⊕ bms 3 .

vir 2 ⊕ deformed bms 3
In the case k = 6, the S-expanded algebra vir D 6 is given by where we have defined The Maxwell-like algebra D 6 [57] is spanned by the generators J 0 , J 1 , J −1 , P 0 , P 1 , P −1 , Z 0 , Z 1 , Z −1 , R 0 , R 1 , R −1 and M 0 , M 1 , M −1 . The algebraic structure of this algebra can be unveiled by performing a suitable change of basis. In fact, two copies of the Virasoro algebra with central charges c = 1 2 (c 4 + c 5 ) andc = 1 2 (c 4 − c 5 ) can be recovered considering the redefinition On the other hand, the change of basiš D -expanded Virasoro algebra vir D 6 is isomorphic to vir 2 ⊕ deformed bms 3 . This procedure can be generalized for higher values of k, showing that vir D k = vir 2 ⊕ vir B k−2 holds generically.

Sugawara construction and expanded Virasoro algebras
The Kač-Moody algebraĝ k corresponds to the central extension of the loop algebra of a semi-simple Lie algebra g and is given by where f abc = −f bac correspond to the structure constants of g and k denotes its central extension.
The Sugawara construction allows one to construct a representation of the Virasoro algebra out of bilinear combinations of the generators of the Kač-Moody algebra by defining ℓ m = 1 2(k + C g ) g ab n : j a n j b m−n : , where C g is the dual Coxeter number of g, g ab is the corresponding Killing-Cartan metric and normal ordering :: is defined as In fact, one can easily check that such definition implies that ℓ m has conformal weight one, [ℓ m , j a n ] = −nj a m+n , and satisfies the Virasoro algebra (2.1) with central charge where dimg =g ab g ab is the dimension of g.

Modified Sugawara construction
The modified Sugawara construction consists in defining new Virasoro generators In the following, we will show how the expanded Virasoro algebras presented in the previous sections can be obtained from Kač-Moody algebras associated with the B k and C k algebras through generalized (modified) Sugawara constructions.

bms 3 and the Sugawara construction
Let us consider the following Kač-Moody like algebra with a semi-direct product structure: which can be obtained from an S-expansion of (6.1) using the semigroup S : j a n p b m−n + p a n j b m−n : − Using the affine current algebra (6.4) it is easy to see that [J m , j a n ] = −nj a m+n , [J m , p a n ] = −np a m+n , [P m , j a n ] = −np a m+n , [P m , p a n ] = 0 , which corresponds to the bms 3 algebra (2.12) with central charges c 1 = 2 dimg and c 2 = 0. The central charge c 1 is familiar from the study of abelian Kač-Moody algebras [1] and manifests here due to the abelian ideal generated by p a m . Now we can use the modified Sugawara construction to obtain the fully centrally extended bm3 3 algebra from (6.4). Indeed defining new generators: This result can be understood as the quantum version of the Sugawara construction described in [22,37] where bms 3 is realized as a Poisson algebra for the central charges of asymptotically flat three-dimensional Einstein gravity.

vir 2 algebra
The bms 3 algebra can also be obtained from the Sugawara construction associated with a Z 2 -expansion of the Kač-Moody algebra, after an IW contraction. In fact, using the semigroup Z 2 = S Redefining the generators as j a m = l a m +l a −m and p a m = l a m −l a −m , this algebra can be written as the product of two identical Kač-Moody algebras with levels k = 1 This means that, considering two independent Sugawara constructions ℓ m = 1 2(k + C g ) g ab n : l a n l b m−n : , ℓ m = 1 2(k + C g ) g ab n :l a nl b m−n : , one can trivially obtain the vir 2 algebra (2.6) with central charges c = kdimg k+Cg andc =k dimḡ k+Cg . Using (2.10), one can define P m = σ 2(k + C g ) g ab n : l a n l b m−n + µl a nl b −m−n : , where µ = k+Cḡ k+Cg . The bilinears (6.8) satisfy the bms 3 algebra (2.12) in the limit σ → ∞ with central charges c 1 = (k−µk) k+Cg dimg and c 2 = (k+µk) k+Cg dimg.

Deformed bms 3 algebra from a Sugawara construction
The Sugawara construction presented before can be generalized in order to recover the deformed bms 3 algebra (3.3) from an expanded Kač-Moody algebra. In this case we introduce the following deformed current algebra: Z m = 1 2k 3 g ab n : z a n z b m−n : , P m = 1 2k 3 g ab n : p a n z b m−n + z a n p b m−n : − J m = 1 2k 3 g ab n : p a n p b m−n + j a n z b m−n + z a n j b m−n : − (6.10) The commutators of J m , P m and Z m with the generators of (6.9) read [J m , j a n ] = −nj a m+n , [P m , j a n ] = −np a m+n , [Z m , j a n ] = −nz a m+n , [J m , p a n ] = −np a m+n , [P m , p a n ] = −nz a m+n , [Z m , p a n ] = 0 , [J m , z a n ] = −nz a m+n , [P m , z a n ] = 0 , [Z m , z a n ] = 0 . In order to obtain the fully centrally extended deformed bms 3 algebra from the deformed affine current algebra (6.9), we introduce the following modified Sugawara construction where the central charges are given bỹ

vir 3 algebra
The deformed bms 3 algebra can also be obtained as an IW contraction of the Sugawara construction associated with an S-expansion of the Kač-Moody algebra using the semigroup where µ = k+Cḡ k+Cg and ν = k+Cg k+Cg . It is easy to verify that the bilinear combinations (6.15) satisfy the deformed bms 3 algebra (3.3) in the limit σ → ∞ with central charges c 1 = (k−µk) k+Cg dimg, c 2 = (k+µk) k+Cg dimg and c 3 = (k+νk) k+Cg dimg.

Generalization
Following the same steps as described above, one can in principle always find a generalized (modified) Sugawara construction that, given a semigroup S, allows one to pass from the Sexpanded Kač-Moody algebra to the corresponding S-expanded Virasoro algebra . As we have seen, the Sugawara construction for the bms 3 algebra and for the deformed bms 3 algebra are quite cumbersome and therefore their generalization for vir B k with k > 4 will not be given here. In the case of the generalized conformal algebras vir C k , the Sugawara constructions presented here have been somewhat straightforward, as the cases k = 3 and k = 4 correspond the direct product of two and three copies of the Virasoro algebra, respectively. However, as we have stressed in Sect. 4.2, for k > 4 it is not true anymore that the vir C k algebras can be written as products of single copies of the Virasoro algebra and therefore the Sugawara construction will be more complicated.

Comments and further developments
In this paper we have presented the general setup to obtain new infinite dimensional algebras by applying the S-expansion method to the Virasoro algebra. Interestingly, the algebras obtained here contain known finite algebras as subalgebras and inherit the way they are related between each other. Indeed, the following diagram summarizes the IW contractions that relate the Poincaré, AdS, Maxwell and AdS-Lorentz algebras in 2 + 1 dimensions as well as their relation with the Lorentz algebra through different S-expansions:

Maxwell
In the first part of this article we have shown that the centrally extended 2D-conformal algebra vir 2 as well as the bms 3 algebra can be obtained as S-expansions of the Virasoro algebra using the semigroups Z 2 and S (1) E , respectively. Subsequently we showed that, using the semigroups S (2) M and S (2) E , the S-expansion leads to three copies of the Virasoro algebra in the former case and to a deformed bms 3 algebra in the latter case. These algebras correspond to infinitedimensional lifts of the AdS-Lorentz and Maxwell algebras and, furthermore, the deformed bms 3 algebra can be obtained as an IW contraction of vir 3 . This means that the infinite dimensional symmetries presented here satisfy the same IW contraction and expansion relations as their finite dimensional subalgebras presented in the previous diagram, i.e., to obtain two sets of families of infinite-dimensional algebras that we have called generalized bms 3 algebras and generalized 2D-conformal algebras. These families are denoted, respectively, by vir C k and vir C k , and reduce to the infinitedimensional algebras previously discussed for k = 3 and k = 4. Furthermore, they turn out to be related by an IW contraction for every value of k Generalized 2D-conformal algebra (vir C k ) In Sect. 5 we have introduced another family of infinite dimensional algebras, vir D k , which can be obtained by expanding the Virasoro algebra using the semigroup S In Sect. 6 the Sugawara construction has been applied to expanded Kač-Moody algebras to obtain the expanded Virasoro algebras and the cases k = 3 and k = 4 have been worked out explicitly. This result is remarkable as it means that these new infinite-dimensional symmetries could be related to some kind of generalized WZW theories whose current algebras are given by expanded Kač-Moody algebras. In that case the algebras vir B k or vir C k should be recovered as the Poisson algebras for the stress-energy momentum tensor components in the very same way as it happens for vir 2 and bms 3 .
In the context of gravity, upon imposing suitable boundary conditions, the algebras vir 2 and bms 3 appear as the asymptotic symmetries of asymptotically AdS and Asymptotically flat three-dimensional Einstein gravity, respectively. We conjecture that the new infinite dimensional algebras vir B k , vir C k and vir D k obtained here correspond the asymptotic symmetries of 3D gravity theories invariant under the algebras B k , C k or D k when suitable boundary conditions for the fields content are adopted. These theories of gravity can be straightforwardly constructed by considering Chern-Simons actions invariant under these algebras. This will be the subject of a subsequent article.
On the other hand, it is well-known that the KdV system possesses a Virasoro symmetry related to the KdV hierarchy [86]. This result can be used to construct an infinite set of boundary conditions for 3D gravity [87]. Along this line it would be interesting to evaluate the existence of integrable systems associated with expanded Virasoro symmetries and they hierarchies as well as their relations to boundary conditions for gravity theories invariant under the algebras B k or C k .
Another natural generalization of our results is to extend the expansion method to Nextended supersymmetric extension of asymptotic symmetries. In particular, it would be interesting to study S-expanded super Virasoro symmetries. However, this would require a more subtle treatment than the one introduced here. Indeed, one cannot naively consider the expansion of a super Virasoro structure. The general setup and the respective supergravity models will be presented in a future paper. As an ending remark: it would be worth exploring the expansion procedure to higher spin extension of gravity theories in 2 + 1 dimensions.

Acknowledgments
This work was supported by the Chilean FONDECYT Projects N • 3170437 (P.C.), N • 3170438 (E.R.) and N • 3160581 (P.S-R.). The authors wish to thank O. Fuentealba, J. Matulich, G. Silva and R. Troncoso for valuable discussion and comments. R.C. would like to thank to the Dirección de Investigación and Vice-rectoría de Investigación of the Universidad Católica de la Santísima Concepción, Chile, for their constant support.
where i takes even values andī takes odd values. This allows one to write the S Let us note that for k = 3, the semigroup corresponds to the S E whose elements satisfy (2.11) and the commutation relations (A.5) reduce to the affine current algebra given by (6.4). The case k = 4 reproduce the S (2) E -expanded algebra whose generators satisfy (6.9). An alternative family of generalized Kač-Moody algebras can be obtained applying the S where {· · · } means the following Interestingly the k = 3 and k = 4 cases reproduce two and three copies of Kač-Moody algebras, respectively. However, for k ≥ 5 the commutation relations of the generalized Kač-Moody algebra obtained here become non-trivial and are given by (A.7). It is important to mention that the two families of generalized Kač-Moody algebras presented here are related through the IW contraction. Indeed, considering the rescaling of the generators satisfyng a S j 0 m → j 0 m , j i m → σ i j i m , pī m → σīpī m , k 1 → k 1 , k i+1 → σ i k i+1 , kī +1 → σīkī +1 the limit σ → ∞ leads to the S (k−2) E -expanded algebra (A.5).