Maxwell Superalgebras and Abelian Semigroup Expansion

The Abelian semigroup expansion is a powerful and simple method to derive new Lie algebras from a given one. Recently it was shown that the $S$-expansion of $\mathfrak{so}\left( 3,2\right) $ leads us to the Maxwell algebra $\mathcal{M}$. In this paper we extend this result to superalgebras, by proving that different choices of abelian semigroups $S$ lead to interesting $D=4$ Maxwell Superalgebras. In particular, the minimal Maxwell superalgebra $s\mathcal{M}$ and the $N$-extended Maxwell superalgebra $s\mathcal{M}^{\left( N\right) }$ recently found by the Maurer Cartan expansion procedure, are derived alternatively as an $S$-expansion of $\mathfrak{osp}\left( 4|N\right) $. Moreover we show that new minimal Maxwell superalgebras type $s\mathcal{M}_{m+2}$ and their $N$-extended generalization can be obtained using the $S$-expansion procedure.


Introduction
The derivation of new Lie algebras from a given one is particularly interesting in Physics since it allows us to find new physical theories from an already known. In fact, an important example consists in obtaining the Poincaré algebra from the Galileo algebra using a deformation process which can be seen as an algebraic prediction of Relativity. At the present, there are at least four different ways to relate new Lie algebras. In particular, the expansion method lead to higher dimensional new Lie algebra from a given one. The expansion procedure was first introduced by Hadsuda and Sakaguchi in Ref. [1] in the context of AdS superstring. An interesting expansion method was proposed by Azcarraga, Izquierdo, Picón and Varela in Ref. [2] and subsequently developed in Refs. [3,4]. This expansion method known as Maurer-Cartan (MC) forms power-series expansion consists in rescaling some group parameters by a factor λ, and then apply an expansion as a power series in λ. This series is truncated in a way that the Maurer-Cartan equations of the new algebra are satisfied.
Another expansion method was proposed by Izaurieta, Rodriguez and Salgado in Ref. [5] which is based on operations performed directly on the algebra generators. This method consists in combining the inner multiplication law of a semigroup S with the structure constants of a Lie algebra g in order to define the Lie bracket of a new algebra G = S × g. This Abelian Semigroup expansion procedure can reproduce all Maurer-Cartan forms power series expansion for a particular choice of a semigroup S. Interestingly, different choices of the semigroup yield to new expanded Lie algebras that cannot be obtained by the MC expansion.
An important property of the S-expansion procedure is that it provides us with an invariant tensor for the S-expanded algebra in terms of an invariant tensor for the original algebra. This is particularly useful in order to construct Chern-Simons and Born-Infeld like actions.
Some examples of algebras obtained as an S-expansion can be found in Refs. [5,6] where the D'auria-Fré superalgebra introduced originally in Ref. [7] and the M algebra are derived alternatively as an S-expansion of osp (32|1). Subsequently, in Refs. [8,9] it was shown that standard odd-dimensional General Relativity can be obtained from Chern-Simons gravity theory for a certain Lie algebra B m and recently it was found that standard even-dimensional General Relativity emerges as a limit of a Born-Infeld like theory invariant under a certain subalgebra of the Lie algebra B m [9,11]. Very recently it was found that the so-called B m Lie algebra correspond to the Maxwell algebras type 1 M m [10]. These Maxwell algebras type M m can be obtained as an S-expansion fo the AdS algebra using S α=0 as an abelian semigroup. The Maxwell algebra has been extensively studied in Refs. [12,13,14,15,16,17,18,19,20,21,22]. This algebra describes the symmetries of a particle moving in a background in the presence of a constant electromagnetic field [12]. Interestingly, in Ref. [17] it was shown that a Maxwell extension of Einstein gravity leads to a generalized cosmological term. Furthermore, it was introduced in Ref. [15] the minimal D = 4 Maxwell superalgebra sM which contains the Maxwell algebra as its bosonic subalgebra. In Ref. [19] the Maurer-Cartan expansion was used in order to obtain the minimal Maxwell superalgebra and its N -extended generalization from the osp (4|N ) superalgebra. This Maxwell superalgebra may be used to obtain the minimal D = 4 pure supergravity from the 2-form curvature associated to sM [21].
The purpose of this work is to show that the abelian semigroup expansion is an alternative expansion method to obtain the Maxwell superalgebra and the N −extended cases. In this way we show that the results of Ref. [19] can be derived alternatively as an S-expansion of the osp (4|N ) superalgebra choos-ing appropriate semigroups. In particular, the minimal Maxwell superalgebra sM is obtained as an S-expansion setting a generator equals to zero. We finally generalize these results proposing new Maxwell superalgebras namely, the minimal Maxwell superalgebras type sM m+2 and the N -extended superalgebras sM (N ) m+2 which can be derived from the osp (4|N ) superalgebra using the S-expansion procedure.
The new sM m+2 superalgebras introduced here are obtained by applying the S-expansion to the osp (4|1) superalgebra and can be seen as supersymmetric extensions of the Maxwell algebras type introduced in Ref. [9]. Unlike the Maxwell superalgebra sM these new superalgebras involve a larger number of extra fermionic generators depending on the value of m. Furthermore, these superalgebras could be used to construct dynamical actions in D = 4 leading to standard supergravity, in a very similar way to the bosonic case considered in Ref. [9].
This work is organized as follows: In section II, we briefly review some aspects of the S-expansion procedure which will be helpful to understand this work. In section III, we review an interesting application of S-expansion procedure in order to get the Maxwell algebra family. Section IV and V contain our main results. In section IV, we present the different minimal D = 4 Maxwell superalgebras which can be obtained from an S-expansion of osp (4|1) superalgebra. In section V, we extend these results to the N -extended case and we present different N -extended D = 4 Maxwell superalgebras from an S-expansion of the osp (4|N ) superalgebra. Section VI concludes the work with a comment about possible developments.

The S-expansion procedure
In this section, we shall review the main aspects of the abelian semigroup expansion method introduce in Ref. [5]. Let us consider a Lie (super)algebra g with basis T A and a finite abelian semigroup S = {λ α }. Then, the direct product G = S × g is also a Lie algebra given by The S-expansion procedure consist in combining the inner multiplication law of a semigroup S with the structure constants of a Lie algebra g. Interestingly, there are differents ways of extracting smaller algebras from G = S × g. But before to extract smaller algebras it is necessary to decompose the original algebra g into a direct sum of subspaces g = p∈I V p , where I is a set of indices. Then for each p, q ∈ I it is possible to define i (p,q) ⊂ I such that Following the definitions of Ref. [5], it is possible to define a subset decomposition S = p∈I S p of the semigroup S such that When such subset decomposition exists, then we say that is a resonant subalgebra of G = S × g. Another case of smaller algebra is when there is a zero element in the semigroup 0 S ∈ S, such that for all λ α ∈ S, we have 0 S λ α = 0 S . In this case, it is possible to reduce the original algebra imposing 0 S × T A = 0 and obtain a new Lie (super)algebra.
Interestingly, there is a way to extract a reduced algebra from a resonant subalgebra. Let G R = p S p × V p be a resonant subalgebra of G = S × g. Let S p =Ŝ p ∪Š p be a partition of the subsets S p ⊂ S such that Then, these conditions induce the decompositioň and therefore Ǧ R corresponds to a reduced algebra of G R . The proofs of these definitions can be found in Ref. [5].
We can see that if we want to obtain an S-expanded algebra, we only need to solve the resonance condition for an abelian semigroup S. In the next section we will briefly review an interesting application of S-expansion procedure in order to derive the Maxwell algebra family. Then we shall see how it is possible to extend this result to the supersymmetric case.

Maxwell algebra as an S-expansion
In order to describe how the S-expansion procedure works, let us review here the results obtained in Refs. [9,10]. The symmetries of a particle moving in a background in presence of a constant electromagnetic field are described by the Maxwell algebra M. This algebra is provided by {J ab , P a , Z ab } where {P a , J ab } do not generate the Poincaré algebra. In fact a particular characteristic of the Maxwell algebra is given by the relation where Z ab commutes with all generators of the algebra except the Lorentz generators J ab , The other commutators of the algebra are Following Refs. [9,10], it is possible to obtain the Maxwell algebra M as an S-expansion of the AdS Lie algebra g using S (2) E as the abelian semigroup. Before to apply the S-expansion procedure it is necessary to consider a decomposition of the original algebra g in subspaces V p , where V 0 is generated by the Lorentz generatorJ ab and V 1 is generated by the AdS boost generatorP a . TheJ ab ,P a generators satisfy the following relations The subspace structure may be written as Let S (2) E = {λ 0 , λ 1 , λ 2 , λ 3 } be an abelian semigroup with the following subset decomposition S (2) E = S 0 ∪ S 1 , where the subsets S 0 , S 1 are given by where λ 3 corresponds to the zero element of the semigroup (0 s = λ 3 ). This subset decomposition is said to be "resonant" because it satisfies [compare with eqs.
In this case, the elements of the semigroup {λ 0 , λ 1 , λ 2 , λ 3 } satisfy the following multiplication law Following the definitions of Ref. [5], after extracting a resonant subalgebra and performing its 0 S -reduction, one finds the Maxwell algebra M = {J ab , P a , Z ab }, whose generators can be written in terms of the original ones, Interestingly, in Refs. [9,11], it was shown that standard four-dimensional General Relativity emerges as a limit of a Born-Infeld theory invariant under a certain subalgebra of the Maxwell algebra M, which was denoted by 2 L M . It was shown that this subalgebra can be obtained as an S-expansion of the Lorentz algebra so (3, 1).
It is possible to extend this procedure and obtain all the possible Maxwell algebras type using the appropriate semigroup.
After extracting a resonant subalgebra and performing its 0 s (= λ 2n )-reduction, one find the Maxwell algebra type 3 M 2n+1 , whose generators are related to the original ones, with i = 0, · · · , n − 1. The commutators of the algebra are We note that setting Z (i+1) ab and Z (i) a equal to zero, we reobtain the Maxwell algebra M. In fact, every Maxwell algebra type M l can be obtained from M 2m+1 setting some bosonic generators equal to zero. These algebras are particularly interesting in gravity context, since it was shown in [9] that standard odd-dimensional general relativity may emerge as the weak coupling constant limit of (2p + 1)-dimensional Chern-Simons Lagrangian invariant under the Maxwell algebra type M 2m+1 , if and only if m ≥ p. Similarly, it was shown that standard even-dimensional general relativity emerges as the weak coupling constant limit of a (2p)-dimensional Born-Infeld type Lagrangian invariant under a subalgebra L M2m of the Maxwell algebra type, if and only if m ≥ p.

S-expansion of the osp (4|1) superalgebra
In this section, we shall take the AdS superalgebra osp (4|1) as a starting point. We will see that different choices of abelian semigroup S lead to new interesting D = 4 superalgebras. In every case, before to apply the S-expansion procedure it is necessary to decompose the original algebra g as a direct sum of where V 0 corresponds to the Lorentz subspace generated byJ ab , V 1 corresponds to the fermionic subspace generated by a 4-component Majorana spinor chargẽ Q α and V 2 corresponds to the AdS boost generated byP a . The osp (4|1) (anti)commutation relations read Here, C stands for the charge conjugation matrix and γ a are Dirac matrices. The subspace structure may be written as The next step consists of finding a subset decomposition of a semigroup S which is "resonant" with respect to (50) − (55).

Minimal D = 4 superMaxwell algebra
Let us consider S (4) E = {λ 0 , λ 1 , λ 2 , λ 3 , λ 4 , λ 5 } as the relevant finite abelian semigroup whose elements are dimensionless and obey the multiplication law In this case, λ 5 plays the role of the zero element of the semigroup S E , so we have for each λ α ∈ S (4) E , λ 5 λ α = λ 5 = 0 s . Let us consider the decomposition One sees that this decomposition is resonant since it satisfies the same structure as the subspaces V p [compare with eqs. (50) − (55)] Following theorem IV.2 of Ref. [5], we can say that the superalgebra is a resonant super subalgebra of S In order to extract a smaller superalgebra from the resonant super subalgebra G R it is necessary to apply the reduction procedure.
Let S p =Ŝ p ∪Š p be a partition of the subsets S p ⊂ S wherě For each p,Ŝ p ∩Š p = ∅, and using the product (56) one sees that the partition satisfies [compare with ecs. (50) − (55)] Then, following definitions of Ref. [5], we havě where and therefore Ǧ R corresponds to a reduced algebra of G R . These S-expansion process can be seen explicitly in the following diagrams: where we have defined J ab,i = λ iJab , P a,i = λ iPa and Q α,i = λ iQα . We can observe that the first diagram corresponds to the resonant subalgebra of the Sexpanded superalgebra S . The second one consists in a particular reduction of the resonant subalgebra.
Thus, the new superalgebra obtained is generated by J ab , P a ,Z ab , Z ab , Q α , Σ α where these new generators can be written as These new generators satisfy the commutation relations where we have used the multiplication law of the semigroup (56) and the commutation relations of the original superalgebra (see Appendix A). The new superalgebra obtained after a reduced resonant S-expansion of osp (4|1) superalgebra corresponds to a generalized minimal superMaxwell algebra sM 4 in D = 4 . One can see that imposingZ ab = 0 leads us to the minimal superMaxwell algebra sM [17,19]. This can be done since the Jacobi identities for spinors generators are satisfied due to the gamma matrix identity (Cγ a ) (αβ (Cγ a ) γδ) = 0 (cyclic permutations of α, β, γ).
In this case, the S-expansion procedure produces a new Majorana spinor charge Σ. The introduction of a second abelian spinorial generator has been initially proposed in Ref. [7] in the context of D = 11 supergravity and subsequently in Ref. [22] in the context of superstring theory.
The sM superalgebra seems particularly interesting in the context of D = 4 supergravity. In fact in Ref. [21], it was shown that D = 4, N = 1 pure supergravity lagrangian can be written as a quadratic expression in the curvatures of the gauge fields associated with the minimal superMaxwell algebra.
It is interesting to note that the expanded superalgebra contains the Maxwell algebra M = {J ab , P a , Z ab } and the Lorentz type subalgebra L M = {J ab , Z ab } introduced in Ref. [11] as subalgebras.

Minimal D = 4 superMaxwell algebra type sM 5
In Ref. [9] , it was shown that the Maxwell algebra type M m can be obtained from an S-expansion of AdS algebra. These bigger algebras require semigroups with more elements but with the same type of multiplication law. Since our main motivation is to obtain a D = 4 superMaxwell algebra type sM m it seems natural to consider a semigroup bigger than S (4) As in the previous case, we shall consider g = osp (4|1) as a starting point with the subspace structure given by eqs. (50) − (55).
Let us consider S } as the relevant finite abelian semigroup whose elements are dimensionless and obey the multiplication law where λ 7 plays the role of the zero element of the semigroup S E . Let us This subset decomposition of S E satisfies the resonance condition since it satisfies the same structure that the subspaces V p [compare with eqs. (50) − (55)] Therefore, according to Theorem IV.2 of Ref. [5], we have that with is a resonant super subalgebra of G = S × g.
As in the previous case, it is possible to extract a smaller superalgebra from the resonant super subalgebra G R using the reduction procedure. Let S p =Ŝ p ∪Š p be a partition of the subsets S p ⊂ S wherě For each p,Ŝ p ∩Š p = ∅, and using the product (91) one sees that the partition satisfies [compare with ecs. (50) − (55)] Then, we haveǦ where and therefore Ǧ R corresponds to a reduced algebra of G R . These procedures can be saw explicitly in the following diagrams: where we have defined J ab,i = λ iJab , P a,i = λ iPa and Q α,i = λ iQα . The first diagram corresponds to the resonant subalgebra of the S-expanded superalgebra S E × osp (4|1). The second one consists in a particular reduction of the resonant subalgebra.
The new superalgebra is generated by J ab , P a , Z ab ,Z ab , Z a ,Z a , Q α , Σ α , Φ α where these new generators can be written as These new generators satisfy the commutation relations where we have used the multiplication law of the semigroup (91) and the commutation relations of the original superalgebra (45) − (49). The new superalgebra obtained after a reduced resonant S-expansion of osp (4|1) superalgebra corresponds to a minimal Maxwell superalgebra type sM 5 in D = 4. Interestingly, this new superalgebra contains the Maxwell algebra type M 5 = {J ab , P a , Z ab , Z a } as a subalgebra [8,9]. In this case, the S-expansion method produces two new Majorana spinors charge Σ and Φ. These fermionic generators transform as spinors under Lorentz transformations. One sees that the minimal superMaxwell type sM 5 requires new bosonic generators Z ab ,Z a , Z a and Σ is not abelian anymore. It is important to note that settingZ ab andZ a equal to zero does not lead to a subalgebra. In fact, these generators are required in Jacobi identity for (Q α , Q β , Σ γ ) due to the gamma matrix identity (Cγ a ) (αβ (Cγ a ) γδ) = Cγ aβ (αβ (Cγ aβ ) γδ) = 0 (cyclic permutations of α, β, γ).
It would be interesting to study this algebraic structure in the context of supergravity theory. It seems that the new minimal Maxwell superalgebra sM 5 defined here may enlarge the D = 4 pure supergravity lagrangian in a particular way.

Minimal D = 4 superMaxwell algebra type sM m+2
Let us further generalize the previous setting. In order to obtain the minimal D = 4 superMawell algebra type sM m+2 , it is necessary to consider a bigger semigroup. Let us consider S (2m) E = {λ 0 , λ 1 , λ 2 , · · · , λ 2m+1 } as the relevant finite abelian semigroup whose elements are dimensionless and obey the multiplication law where λ 2m+1 plays the role of the zero element of the semigroup. Let us consider the decomposition S where the subsets S 0 , S 1 , S 2 are given by the general expression This decomposition is said to be resonant since it satisfies [compare with eqs. (50) − (55)] Therefore, we have with is a resonant subalgebra of G = S × g. As in previous cases, it is possible to extract a smaller algebra from the resonant subalgebra G R using the reduction procedure. Let S p =Ŝ p ∪Š p be a partition of the subsets S p ⊂ S wherě where (λ 2m ) means that λ 2m ∈Ŝ 0 if m is odd and λ 2m ∈Ŝ 2 if m is even. For each p,Ŝ p ∩Š p = ∅, and using the product (126) one sees that the partition satisfies [compare with ecs. (50) − (55)] ThereforeǦ corresponds to a reduced algebra of G R , wherě Here,J ab ,P a andQ α correspond to the generators of osp (4|1) superalgebra. The new superalgebra obtained by the S-expansion procedure is generated by where these new generators can be written as . It is important to note that the super indices k and l of spinor generators correspond to the expansion labels and they do not define an N -extended superalgebra. The N -extended case will be considered in the next section.

2
. The commutation relations can be obtained using the multiplication law of the semigroup (126) and the commutation relations of the original superalgebra (45) − (49). One sees that when k + l > m 2 then the generatos T  [8,9]. Interestingly, when m = 2 and imposingZ (1) ab = 0 we recover the minimal Maxwell superalgebra sM. The case m = 1 corresponds to D = 4 Poincaré superalgebra sP = {J ab , P a , Q α }. This is not a surprise since the reduced resonant S In this case, the S-expansion method produces new Majorana spinors charge Σ (k) and Φ (l) . These fermionic generators transform as spinors under Lorentz transformations. One can see that the Jacobi identities for spinors generators are satisfied due to the gamma matrix identity (Cγ a ) (αβ (Cγ a ) γδ) = Cγ aβ (αβ (Cγ aβ ) γδ) = 0 (cyclic permutations of α, β, γ). In fact, all the commutators satisfy the JI since they correspond to expansions of the original JI of osp (4|1).

N-extended superMaxwell algebras
We have shown that the minimal D = 4 Maxwell superalgebras type sM m+2 can be obtained from a reduced resonant S (2m) E -expansion of osp (4|1) superalgebra. It seems natural to expect to obtain the D = 4 N -extended Maxwell superalgebras from an S-expansion of osp (4|N ) superalgebra.
If we want to apply an S-expansion, first it is convenient to decompose the original superalgebra g as a direct sum of subspaces V p , where V 0 corresponds to the subspace generated by Lorentz generatorsJ ab and by N (N −1) 2 internal symmetry generators T ij , V 1 corresponds to the fermionic subspace generated by N Majorana spinor chargesQ i α (i = 1, · · · , N ; α = 1, · · · , 4) and V 2 corresponds to the AdS boost generated byP a . The osp (4|N ) (anti)commutation relations read where i, j, k, l = 1, . . . , N . The subspace structure may be written as Let us consider S (4) E = {λ 0 , λ 1 , λ 2 , λ 3 , λ 4 , λ 5 } as the relevant finite abelian semigroup whose elements are dimensionless and obey the multiplication law In this case, λ 5 plays the role of the zero element of the semigroup S This subset decomposition satisfies the resonance condition since we have [compare with eqs. (180) − (185)] Thus, according to Theorem IV.2 of Ref. [5], we have that is a resonant subalgebra of S (4) Imposing λ 5 T A = 0, the 0 S -reduced resonant superalgebra is obtained. The new superalgebra is generated by J ab , P a , Z ab ,Z ab ,Z a , Q i α , Σ i α , T ij , Y ij ,Ỹ ij where the new generators can be written as Then using the multiplication law of the semigroup (186) and the commutations relations of the original superalgebra (173) − (179) it is possible to write the resulting superalgebra as The new superalgebra obtained after a reduced resonant S 4 . An alternative expansion procedure to obtain the N -extended Maxwell superalgebra has been proposed in Ref. [19]. Interestingly, this superalgebra contains the generalized Maxwell algebra gM = J ab , P a , Z ab ,Z ab ,Z a as a subalgebra (see Appendix B). One sees that the S-expansion procedure introduces additional bosonic generators which modify the minimal Maxwell superalgebra [see eqs. (212), (213)]. Naturally whenZ a =Z ab = Y ij =Ỹ ij = 0, we obtain the simplest D = 4 N -extended Maxwell superalgebra sM (N ) generated by J ab , P a , Z ab , Q i α , Σ i α , T ab . Eventually for N = 1, with T ab = 0, the D = 4 minimal Maxwell superalgebra sM is recovered. It is important to note that setting some generators equals to zero does not always lead to a Lie superalgebra. Nevertheless, the properties of the gamma matrices in 4 dimensions permit us to impose some generators equals to zero without breaking the Jacobi Identity.
We can generalize this procedure and obtain the N -extended superMaxwell algebra type sM (N ) m+2 as an reduced resonant S-expansion of osp (4|N ) with S (2m) E = {λ 0 , λ 1 , λ 2 , · · · , λ 2m+1 } as abelian semigroup. In fact, if we consider a resonant subset decomposition S and let S p =Ŝ p ∪Š p be a partition of the subsets S p ⊂ S wherě where (λ 2m ) means that λ 2m ∈Ŝ 0 if m is odd and λ 2m ∈Ŝ 2 if m is even. This decomposition satifies the resonant condition for any value of m and we find thatǦ corresponds to a reduced resonant algebra. This new superalgebra correspond to the N -extended Maxwell superalgebra type sM (N ) m+2 which is generated by These generators can be written as The commutation relations can be obtained using the multiplication law of the semigroup and the commutation relations of the osp (4|N ) superalgebra. As in the case of minimal superMaxwell algebra type one sees that when k+q > m 2 then the generators T where i, j, g, h = 1, · · · , N . Naturally, when k + j > m then the generators T B are abelian. With this notation it is not trivial to see the Maxwell algebra type M m+2 as a subalgebra. However it could be useful in order to construct an action for this superalgebra.

Comments and possible developments
In the present work we have shown that the Maxwell superalgebras found by the MC expansion method in Ref. [19] can be derived alternatively by the S-expansion procedure. In particular, the S-expansion of osp (4|1) permits us to obtain the minimal Maxwell superalgebra sM. Then choosing different semigroups we have shown that it is possible to define new minimal D = 4 Maxwell superalgebras type sM m+2 which can be seen as a generalization of the D'Auria-Fré superalgebra and the Green algebras introduced in Refs. [7], [22], respectively. Interestingly, the case m = 1 corresponds to the minimal Poincaré superalgebra. Recently it was shown that the minimal Maxwell superalgebra sM may be used to obtain the minimal D = 4 pure supergravity [21]. It seems that the new minimal Maxwell superalgebras sM m+2 defined here may be good candidates to enlarge the D = 4 pure supergravity lagrangian leading to a generalized cosmological term. Interestingly we have shown that this Maxwell superalgebra contains the Maxwell algebras type M m+2 as bosonic subalgebras.
We also have shown that the D = 4 N -extended Maxwell superalgebra sM (N ) , derived initially as a MC expansion in Ref. [19], can be obtained alternatively as an S-expansion of osp (4|N ). In this case the S-expansion produces additional bosonic generators which modify the minimal Maxwell superalgebra. Choosing bigger semigroups we have shown that it is possible to define new D = 4 N -extended Maxwell superalgebras type sM (N ) m+2 . Naturally when m = 2 we recover the sM (N ) superalgebra and for N = 1 we recover the Maxwell algebra type sM m+2 . It would be interesting to build lagrangians with the 2-form curvature associated to these new N -extended Maxwell superalgebras sM Thus, we have shown that the S-expansion procedure is a powerful and simple tool in order to derive new Lie superalgebras. In fact, the introduction of new Majorana spinor charges could not be guessed trivially. The method considered here could play an important role in the context of supergravity in higher dimensions. It seems that it should be possible to recover standard oddand even-dimensional supergravity from the Maxwell superalgebra family [work in progress].

A S-expansion of the commutation relations
Let g be a Lie (super)algebra given by Let S = {λ α } be an abelian semigroup with 2-selector K γ αβ . Let us denote a basis element of the direct product S × g by T (A,α) = λ α T A and consider the induced commutator T (A,α) , T (B,β) = λ α λ β [T A , T B ]. Then S × g is also a Lie (super)algebra with structure constants We have said that a decomposition of the original algebra g is given by, Let S p =Ŝ p ∪Š p be a partition of the subsets S p ⊂ S Then, we have said thať corresponds to a reduced resonant superalgebra. Thus the new Majorana spinor charges are given by whereQ α corresponds to the original Majorana spinor charge. Then, the new anticommutators are given by where we have used thatZ ab = J ab,2 = λ 2Jab and P a = P a,2 = λ 2Pa . In the same way, it is possible to show that where we have used that Z ab = J ab,4 = λ 4Jab . This procedure can be extended to any (anti)commutator of a S-expanded (super)algebra.

B Generalized Maxwell algebra in D = 4 as an S-expansion
In this appendix we will show how to obtain the generalized Maxwell algebra gM from so (3, 2) using the S-expansion procedure.
Here λ 3 plays the role of the zero element of the semigroup S E . Let us consider a subset decomposition S This subset decomposition is said to be "resonant" because it satisfies [compare with eqs.(260) − (262).] Imposing the 0 S -reduction condition, we find a new Lie algebra generated by J ab , P a , Z ab ,Z ab ,Z a where we have defined J ab = J ab,0 = λ 0Jab , P a = P a,1 = λ 1Pa , Z ab = J ab,2 = λ 2Jab , Z a = P a,2 = λ 2Pa .
These new generators satisfy the commutation relations [J ab , J cd ] = η bc J ad − η ac J bd − η bd J ac + η ad J bc , [J ab , P c ] = η bc P a − η ac P b , [J ab , Z cd ] = η bc Z ad − η ac Z bd − η bd Z ac + η ad Z bc , J ab ,Z cd = η bcZad − η acZbd − η bdZac + η adZbc , Z ab ,Z cd = η bc Z ad − η ac Z bd − η bd Z ac + η ad Z bc , J ab ,Z c = η bcZa − η acZb , Z ab , P c = η bcZa − η acZb , where we have used the multiplication law of the semigroup (263) and the commutation relations of the original algebra. The new algebra obtained after a 0 Sreduced resonant S-expansion of so (3, 2) corresponds to a generalized Maxwell algebra gM in D = 4 [19]. This new algebra contains the Maxwell algebra M as a subalgebra. It is interesting to observe that the gM algebra is very similar to the Maxwell algebra type M 6 introduced in Refs. [8,9]. In fact, one could identify Z ab ,Z ab andZ a with Z ab , Z ab and Z a of M 6 respectively. However, the commutation relations (277), (280) and (282) are subtly different of those of Maxwell algebra type M 6 .