Generalization of extended Lie algebras by expansions of extended de Sitter algebra, in four-dimensions

TL;DR

This paper explores how to generate extended Lie algebras, including Poincaré and AdS-Lorentz, through algebra expansion methods from de Sitter and conformal algebras, introducing new algebra families and their flat limits.

Contribution

It generalizes the expansion procedure to create new extended algebra families and demonstrates their relation to known algebras via contractions.

Findings

New extended algebra families $ ext{C}_k^{E}$ and $ ext{B}_k^{E}$ are constructed.

Extended Poincaré algebra obtained as a contraction of extended de Sitter algebra.

The expansion method applies to four-dimensional spacetime algebras.

Abstract

Four-dimensional extended: Poincar\'e, AdS-Lorentz and Maxwell algebras, are obtained by expanding an extension of de Sitter or conformal algebra, SO(4,1) or SO(3,2). The procedure can be generalized to obtain a new family of extended $\mathcal{C}_k^{E}$ and its flat limit, the extended $\mathcal{B}_k^{E}$ algebras. The extended $\mathcal{C}_k$ and $\mathcal{B}_k$ algebras have been introduced in the literature recently.The extended Poincar\'e algebra is also obtained as an In\"on\"u-Wigner contraction of extended de Sitter algebra.

Equations

$$\left[J^{AB},J^{CD}\right]=i(g^{AD}{J}^{BC}-g^{AC}{J}^{BD}+g^{BC}{J}^{AD}-g^{BD}{J}^{AC})$$

$$Ad{S}^E=Span\left\{J^{AB},L_a^{C_1...C_s}\right\},$$

$$\left[J^{AB},J^{CD}\right]=i(g^{AD}{J}^{BC}-g^{AC}{J}^{BD}+g^{BC}{J}^{AD}-g^{BD}{J}^{AC})$$

$$\left[J^{AB},L_a^{C_1...C_s}\right]=i({\eta }^{C_{1}B}{L}_a^{A{C}_2...C_s}-...-{\eta }^{C_{s}A}{L}_a^{C_1...C_{s-1}B})$$

$$\left[L_a^{C_1...C_n},L_b^{C_{n+1}...C_s}\right]=i{f}_{abc}{L}_c^{C_1...C_s}\left(s=0,1,2,....\right),$$

$$\begin{array}[]{ccc}L_{a}^{\lambda_{1}...\lambda_{s}}=e^{\lambda_{1}}....e^{\lambda_{s}}\otimes L_{a}&,&s=0,1,2,...\end{array}$$

$$\begin{array}[]{ccc}\left[L_{a},L_{b}\right]=if_{abc}L_{c}&,&a,b,c=1,...,\dim L_{G}.\end{array}$$

$$\begin{array}[]{cccc}J^{4\mu}:=P^{\mu}&,&M^{\mu\nu}:=J^{\mu\nu}&,\end{array}$$

$${\mathcal{C}}_3^E=Span\left\{P^{\mu },J^{\mu \nu },L_a^{C_1...C_s}\right\}$$

$$\left[P^{\mu },P^{\nu }\right]$$

$$\left[J^{\mu \nu },J^{\rho \sigma }\right]$$

$$\left[J^{\nu \rho },P^{\mu }\right]$$

$$\left[J^{\mu \nu },L_a^{C_1...C_s}\right]=i({\eta }^{C_1\nu }{L}_a^{\mu C_2...C_s}-...-{\eta }^{C_s\mu }{L}_a^{C_1...C_{s-1}\nu }),$$

$$\left[P^{\mu },L_a^{C_1...C_s}\right]=i({\eta }^{C_1\mu }{L}_a^{4C_2...C_s}-...-{\eta }^{C_{s}4}{L}_a^{C_1...C_{s-1}\mu }),$$

$$\left[L_a^{C_1...C_n},L_b^{C_{n+1}...C_s}\right]=i{f}_{abc}{L}_c^{C_1...C_s}\left(s=0,1,2,....\right).$$

$$\begin{array}[]{ccc}V_{0}=\left\{J^{\mu\nu},L_{a}^{C_{1}...C_{s}}\right\}&and&V_{1}=\left\{P^{\mu}\right\}\end{array}$$

$$\left[{\mathcal{P}}^{\mu },{\mathcal{P}}^{\nu }\right]$$

$$\left[{\mathcal{J}}^{\mu \nu },{\mathcal{P}}^{\rho }\right]$$

$$\left[{\mathcal{J}}^{\mu \nu },{\mathcal{J}}^{\rho \sigma }\right]$$

$$\left[{\mathcal{P}}^{\mu },{\mathcal{L}}_a^{C_1...C_s}\right]$$

$$\left[{\mathcal{J}}^{\mu \nu },{\mathcal{L}}_a^{C_1...C_s}\right]$$

$$\left[{\mathcal{L}}_a^{C_1...C_n},{\mathcal{L}}_b^{C_{n+1}...C_s}\right]$$

$$\begin{array}[]{ccc}S_{0}=\left\{\lambda_{0},\lambda_{2}\right\}&,&S_{1}=\left\{\lambda_{1},\lambda_{2}\right\},\end{array}$$

$$\begin{array}[]{cccc}\ast&\lambda_{0}&\lambda_{1}&\lambda_{2}\\ \lambda_{0}&\left\{\lambda_{0},\lambda_{2}\right\}&\left\{\lambda_{1},\lambda_{2}\right\}&\lambda_{2}\\ \lambda_{1}&\left\{\lambda_{1},\lambda_{2}\right\}&\left\{\lambda_{0},\lambda_{2}\right\}&\lambda_{2}\\ \lambda_{2}&\lambda_{2}&\lambda_{2}&\lambda_{2}\end{array}$$

$$S^{(3)}=\begin{array}[]{cccc}\ast&\lambda_{0}&\lambda_{1}&\lambda_{2}\\ \lambda_{0}&\lambda_{0}&\lambda_{1}&\lambda_{2}\\ \lambda_{1}&\lambda_{1}&\lambda_{2}&\lambda_{2}\\ \lambda_{2}&\lambda_{2}&\lambda_{2}&\lambda_{2}\end{array}.$$

$$W_0=S_0\times V_0=\left\{{\lambda }_0{J}^{\mu \nu },{\lambda }_0{L}_a^{C_1...C_n},{\lambda }_2{J}^{\mu \nu },{\lambda }_2{L}_a^{C_1...C_n}\right\}$$

$$W_1=S_1\times V_1=\left\{{\lambda }_1{P}^{\mu },{\lambda }_2{P}^{\mu }\right\}$$

$${\mathcal{B}}_R=W_0\oplus W_1=\left\{{\lambda }_0{J}^{\mu \nu },{\lambda }_0{L}_a^{C_1...C_n},{\lambda }_2{J}^{\mu \nu },{\lambda }_2{L}_a^{C_1...C_n},{\lambda }_1{P}^{\mu },{\lambda }_2{P}^{\mu }\right\}.$$

$${\mathcal{B}}_{R0}=\left\{{\lambda }_0{J}^{\mu \nu },{\lambda }_0{L}_a^{C_1...C_n},{\lambda }_1{P}^{\mu }\right\}.$$

$$\begin{array}[]{ccc}\mathcal{J}^{\mu\nu}=\lambda_{0}J^{\mu\nu},&l\mathcal{P}^{\mu}=\lambda_{1}P^{\mu},&\mathcal{L}_{a}^{C_{1}...C_{n}}=\lambda_{0}L_{a}^{C_{1}...C_{n}},\end{array}$$

$$S^{(4)}=\begin{array}[]{ccccc}\ast&\lambda_{0}&\lambda_{1}&\lambda_{2}&\lambda_{3}\\ \lambda_{0}&\lambda_{0}&\lambda_{1}&\lambda_{2}&\lambda_{3}\\ \lambda_{1}&\lambda_{1}&\lambda_{2}&\lambda_{1}&\lambda_{3}\\ \lambda_{2}&\lambda_{2}&\lambda_{1}&\lambda_{2}&\lambda_{3}\\ \lambda_{3}&\lambda_{3}&\lambda_{3}&\lambda_{3}&\lambda_{3}\end{array}$$

$$\begin{array}[]{ccc}S_{0}=\left\{\lambda_{0},\lambda_{2},\lambda_{3}\right\}&,&S_{1}=\left\{\lambda_{1},\lambda_{3}\right\},\end{array}$$

$$W_0=S_0\times V_0=\left\{{\lambda }_0{J}^{\mu \nu },{\lambda }_0{L}_a^{C_1...C_n},{\lambda }_2{J}^{\mu \nu },{\lambda }_2{L}_a^{C_1...C_n},{\lambda }_3{J}^{\mu \nu },{\lambda }_3{L}_a^{C_1...C_n}\right\},$$

$$W_1=S_1\times V_1=\left\{{\lambda }_1{P}^{\mu },{\lambda }_3{P}^{\mu }\right\},$$

Full text

Generalization of extended Lie algebras by expansions of extended de Sitter algebra, in four dimensions.

Ricardo Caroca

*Departamento de Matemática y Física Aplicadas, *

• Universidad Católica de la Santísima Concepción, *

Alonso de Ribera 2850, Concepción, Chile.

rcaroca@uscs.cl

Four-dimensional extended: Poincaré, AdS-Lorentz and Maxwell algebras, are obtained by expanding an extension of de Sitter or conformal algebra, SO(4,1) or SO(3,2). The procedure can be generalized to obtain a new family of extended $\mathcal{C}_{k}^{E}$ and its flat limit, the extended $\mathcal{B}_{k}^{E}$ algebras. The extended $\mathcal{C}_{k}$ and $\mathcal{B}_{k}$ algebras have been introduced in the literature recently.The extended Poincaré algebra is also obtained as an Inönü-Wigner contraction of extended de Sitter algebra.

1 Introduction

One of the great challenges of contemporary physics is to reconcile quantum mechanics with general relativity, the two last and greatest conceptual revolutions of physics, to describe the four fundamental interactions in a single theoretical framework, in addition to solving a series of phenomenological problems that exist today, it’s still an open problem [1].

The supersymmetry and the supergravity theories may remove some ultraviolet divergences in perturbation theory, which is one of the big problems today, but the problem with supergravity theory is that the restriction $s\leq 2$ on the spins of the particles in the massless supergravity supermultiplet leads to a famous limitation $N\leq 8$ on the number of gravitinos. The gravitino is a massless spin-two field and it is the quantum mechanical particle that carries gravitational force and therefore plays a crucial role, showing the inability of the supergravity to work with interacting higher-spin gauge fields [2].  For its part, the string theories that involve fields of arbitrary spin (higher-spin), the ultraviolet divergence is solved because the point-like particles of the Standard Model are replaced by extended objects called strings. Fundamental properties such as: mass, charge and other, are determined by the vibrational state of the string, and one of the vibrational states gives rise to the graviton. Therefore, the string theory incorporates all of the fundamental interactions, including gravity and the superstring theory describe infinite collections of higher-spin excitations of all massive spins [4], [5].  Five consistent versions ended being different limits of the M-theory in eleven dimensions. Lamentably, there are no clear observable predictions in the experiments, due to the high energies they require, besides having some unsolved problems such as the compactified dimensions and some mathematical problems. Therefore, it is natural to ask the following question: is there a certain spontaneous symmetry breaking mechanism from some underlies symmetric phase [2], to explain the massive excitations of string theories ?. There is gauge theories of interacting fields of all spin that are based in infinite-dimensional gauge symmetries, and its are know as higher-spin gauge theories (HS) [2], [6]-[12], which can be considered as most general gauge theories in (3+1) space-time dimensions. The higher-spin gauge symmetries describe infinite collection of interacting higher-spin massless field of all spin $0\leq s\leq\infty$ [6]-[12] and notably the HS theories contain lower-spin gauge symmetries ($s\leq 2$) as finite-dimensional subalgebras [2].

On the other hand, a higher-spin gauge extension coupled to the three-dimensional AdS gravity was formulated as a Chern-Simons theory [3] whose gauge group is given by $SL(n)\times SL(n)$ and two copies of $\mathcal{W}$-algebras as asymptotic symmetries of asymptotically Anti-de Sitter solutions was obtained.  The particular case of a spin-3 field for the group $SL(3)\times SL(3$) was studied in detail in Ref. [3] and two copies of the classical $\mathcal{W}_{3}$-algebra with central charge was resulted, describing the coupling of a spin-3 gauge field to gravity.

Another interesting point of view is to extend the Poincaré group, introdu-cing fermionic generators with half-integer spin generators, in three space-time dimensions. It was shown that hypergravity could be formulated as a local gauge symmetry [13], [14]. An Chern-Simons action invariant to the extension of the Poincaré group can be formulated, since this algebra admits a nontrivial Casimir operator. An hypergravity in the generic case of massless fermionic field, could be formulated, to the case of hyper-Poincaré algebra.

Another way to approach to the higher-spin gauge symmetries, is the proposal to extend the (Super)Poincaré group, and consists of introducing a mix of internal and space-time symmetries [15], [16], [17], in four space-time dimensions. Starting from the base generators of an internal compact Lie algebra, it consists of introducing an infinite set of translationally invariant generators, which are totally symmetrical with respect to the space-time indices. These generators close as subalgebra in the extended Poincaré algebra, are s-range tensors and have a nonzero commutation relation with the Lorentz generators, and therefore are higher spin carriers. The massless case and the irreducible representations of the extended symmetry were constructed. The transversal representations imply an infinite series of helicities of integers and semi-integers.

An extension of the de Sitter and conformal group with the same approach is suggested, and will be the starting point of this work [16]-[21]. We can say that the construction of consistent ”higher-spin gauge theory”, suggests that the algebra that supports it must be infinite-dimensional. In what follows, we will use the classification which have been recently introduced in the literature in the context of: (super)gravity [22], [23], higher-spin gravity [24] and the asymptotic symmetries [25]; to designate the generalizations of the families algebras: the AdS ($\mathcal{C}_{3}$) and Poincaré ($\mathcal{B}_{3}$), AdS$\oplus$Lorentz ($\mathcal{C}_{4}$) and Maxwell ($\mathcal{B}_{4}$), and therefore the generaliced $\mathcal{C}_{k}$ and $\mathcal{B}_{k}$ algebras [26], [27], [28].

The paper is organized as follows: In the section 2 we briefly review an extension of de Sitter and conformal algebras ($\mathcal{C}_{3}^{E}$) and subsequently, in section 2.1, we show that extended Poincaré algebra ($\mathcal{B}_{3}^{E}$) can be obtained as a Inönü-Wigner contraction of extended de Sitter ($\mathcal{C}_{3}^{E})$ algebra. The section 3 is devoted entirely to the results obtained by applying the S-expansion method to extended de Sitter algebra ($\mathcal{C}_{3}^{E}$). In chapter 3.1, the extended Poincaré Algebra ($\mathcal{B}_{3}^{E}$) as a resonant and reduced subalgebra of $\mathcal{C}_{3}^{E}$, was obtained. Chapter 3.2 we construct the commutations relations of extended AdS$\oplus$Lorentz ($\mathcal{C}_{4}^{E}$) through a reduced subalgebra of extended de Sitter algebra ($\mathcal{C}_{3}^{E}$). Subsequently, in section 3.3 by extending a flat limit we rescued the extended Maxwell algebra ($\mathcal{B}_{4}^{E}$) from extended AdS$\oplus$Lorentz. In the section 3.4 the previous results are generalized, where extended $\mathcal{C}_{k}^{E}$ algebra as a reduced subalgebra of extended de Sitter ($\mathcal{C}_{3}^{E}$) was obtained. In the section 3.4 we found the extended $\mathcal{B}_{k}^{E}$ algebra as a flat limit of extended $\mathcal{C}_{k}^{E}$ algebra. Finally, we conclude with some comments and possible future developments.

2 An extension of de Sitter and conformal algebras ($\mathcal{C}_{3}^{E}$).

Consider first the de Sitter and conformal algebras $SO(4,1)$ or $SO(3,2)$, proposed by [16] (section VI, equation (39)), whose generators satisfy the commutation relation

[TABLE]

where $g^{AB}=\left(+----\right)$ or $g^{AB}=\left(+---+\right)$  and $A,B=0,1,..,4$.

According to to [16], [17], [18], [19], [20],[21] we might postulate an extension for the de Sitter and conformal groups.  For this, consider the extended de Sitter algebra in 4-dimensions, $\mathcal{A}d\mathcal{S}^{E}$, whose generators are

[TABLE]

and the sets of commutators are the following [16].

[TABLE]

[TABLE]

[TABLE]

where the infinite set of generators $L_{a}^{\lambda_{1}...\lambda_{s}}$ is defined, in the form

[TABLE]

The generators $L_{a}^{\lambda_{1}...\lambda_{s}}$ carry internal and space-time indices, and are totally symmetric with respect to the indices $\lambda_{1}...\lambda_{s}$ [16]. The algebra (2)-(4) corresponds to a generalization to the cases of the de Sitter groups.

On the other hand, the generators $L_{a}$ correspond to the internal compact Lie algebra $L_{G}$,

[TABLE]

where the internal bosonic algebra obeys the Coleman-Mandula theorem.

We can check that all Jacoby identities are satisfied and we have an consistent algebra (2)-(4).

2.1 Extended Poincaré Algebra ($\mathcal{B}_{3}^{E}$) as a Inönü-Wigner contraction of extended de Sitter ($\mathcal{C}_{3}^{E})$ algebra.

We rewrite the generators of $\mathcal{C}_{3}^{E}$ algebra (2)-(4) fixing some indices as follows:

[TABLE]

where $\mu,\nu,\rho=0,1,2,3$.  Then, the extended (anti) Sitter algebra, $\mathcal{A}d\mathcal{S}^{E}$( $SO(3,2)$ ) in 5-dimensions, written in terms of the new generators,

[TABLE]

where $sig(g^{\mu\nu})=\left(+---\right)$, has the following commutations relations:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The $\mathcal{C}_{3}^{E}$ algebra (9)-(14) presents a structure of subalgebra $V_{0}$ and symmetric coset $V_{1}$ (88); where

[TABLE]

Making a Inönü-Wigner contraction [29], [30], [31], [32]: $P^{\mu}\rightarrow lP^{\mu}$andand extending the flat limit $l\longrightarrow\infty$, reduces it to the Extended Poincaré algebra ($\mathcal{B}_{3}^{E}$); that is to say:

[TABLE]

Note that because the generators $\mathcal{L}_{a}^{C_{1}...C_{n}}$ have a nonzero commutations relation with the generator $\mathcal{J}^{\mu\nu}$, they carry higher spins gauge fields. In this case, the infinite set of generators $L_{a}^{\lambda_{1}...\lambda_{s}}$ are translationally invariant (19).

The first three commutations relations correspond to the usual Poincaré algebra ($\mathcal{B}_{3}$) generated by $\left\{\mathcal{P}^{\mu},\mathcal{J}^{\mu\nu}\right\}$, while the last three commutations relations describe the coupling of the higher-spin generators $\left\{\mathcal{L}_{a}^{C_{1}...C_{n}}\right\}$ to the Poincaré symmetry.

This last result (16)-(21) corresponds the the same result obtained in (9), (10) and (11) of reference [16].

3 S-expansions of extended de Sitter algebra ($\mathcal{C}_{3}^{E}$).

In this section we will use the S-expansion procedure of Lie algebras [33], [34], which is briefly shown in Appendices A and B, to obtain: extended Poincaré, extended AdS$\oplus$Lorentz, extended Maxwell, and generalized extended $\mathcal{C}_{k}^{E}$ and generalized extended $\mathcal{B}_{k}^{E}$ algebras. The $C_{k}$ is a semisimple Lie algebras (AdS ”type”) being a generalization that contains the AdS and AdS$\oplus$Lorentz algebras, as particular cases. In contrast, $B_{k}$ is a nonsemisimple Lie algebras (Poincaré ”type”) and are generalizations that contains the Poincaré and Maxwell algebras as particular cases. It is important to note also that all Jacobi identities (87) are guaranteed for expanded algebras due to the associative and commutative property of the semigroup in the S-expansion procedure, [33], [34]. Therefore, all the expanded Lie algebras are closed algebras, and have explicit matrix representations.

3.1 Extended Poincaré Algebra ($\mathcal{B}_{3}^{E}$) as a resonant

and reduced subalgebra of $\mathcal{C}_{3}^{E}$.

Using the S-expansion method of Lie algebras [33]-[36], shown in summary form in the appendices A and B, and let us consider the set $\ S^{(3)}=\left\{\lambda_{0},\lambda_{1},\lambda_{2}\right\}$, with the resonant partition (89)

[TABLE]

to the resonant conditions (88) to the subspaces (15). There are $2^{4}=16$ different sets that satisfy the resonant condition (88), and these have the following multiplication rules

[TABLE]

and for example, the set $\left\{\lambda_{0},\lambda_{2}\right\}$ in the table means the product $\lambda_{0}\ast\lambda_{0}$ can be $\lambda_{0}$ or $\lambda_{2}.$ From the multiplication table (23), we choose the following semigroup

[TABLE]

Following the S-expansion method discussed in Appendices A and B to expand the extended de Sitter algebra, $\mathcal{A}d\mathcal{S}^{E}$ (9)-(14), using the semigroup (24), considering the partition in subspaces (15) y (88), and the resonant partition (22), we have the resonating subspaces (91)

[TABLE]

[TABLE]

and the resonant subalgebra (90),

[TABLE]

Choosing the zero element $\lambda_{2}=0_{S}$ of the semigroup, we obtain the reduced resonant subalgebra (92)

[TABLE]

We redefine the previous generators in the form

[TABLE]

using the extended de Sitter algebra, $\mathcal{A}d\mathcal{S}^{E}$ (9)-(14), the semigroup (24), and extending the flat limit $l\longrightarrow\infty$; we obtain the extended Poincaré algebra ($\mathcal{B}_{3}^{E}$), (16)-(21), that corresponds to the same result obtained in (9), (10) and (11) of reference [16].

3.2 Extended AdS$\oplus$Lorentz ($\mathcal{C}_{4}^{E}$) as a

resonant and reduced subalgebra of extended de Sitter algebra ($\mathcal{C}_{3}^{E}$).

To obtain extended AdS$\oplus$Lorentz ($\mathcal{C}_{4}^{E}$) [22], [23], [24], [25], we should consider the semigroup $S^{(4)}=\left\{\lambda_{0},\lambda_{1},\lambda_{2},\lambda_{3}\right\}$, whose multiplication table is the following

[TABLE]

(note that the table (24) is embedding in the table (30)), with the partition of the semigroup

[TABLE]

which satisfies the resonant condition (89).

Following the S-expansion method [33]-[36] of Appendices A and B, using the subspaces (15), (88) and the decomposition (31), we have the resonating subspaces (91)

[TABLE]

and

[TABLE]

also the corresponding resonant subalgebra (90),

[TABLE]

Defining $\lambda_{3}=0_{S}$ as the zero of the semigroup, the resonant and reduced subalgebra is the following

[TABLE]

We must impose an extra condition between some elements of the semigroup, for the algebra to be closed, that is to say,

[TABLE]

therefore the semigroup (30) is reduced to the semigroup

[TABLE]

($\mathcal{S}_{red.}^{(4)}$ does not lose the associative property due to the (36) condition),

Taking into account the condition (36), the semigroup (37) continues to satisfy the resonant condition (89) and we define the generators of (35) in the form

[TABLE]

Using the generators (41) and the semigroup (37), we get by expansion the extension of the AdS$\oplus$Lorentz algebra ($\mathcal{C}_{4}^{E}$) from the extension of de Sitter algebra (9)-(14):

[TABLE]

[TABLE]

Note that in this case the generators \left\{\begin{array}[]{cc}\mathcal{L}_{ai}^{C_{1}...C_{s}},&i=0,1\end{array}\right\} carry higher-spins gauge fields and differently from the case of the Poincaré algebra, here the higher-spins generators are not translationally invariant, because the commutation relation (13) is not null. The algebra (42)-(43) satisfy all Jacoby identities (87) because it is an expansion (see apppendix A) of the extended algebra (9)-(14) and the latter is only obtained by fixing some indexes of extended de Sitter algebra (2)-(4).

The first six commutators (42) correspond to the AdS$\oplus$Lorentz algebra and the remaining (43) correspond to the extended sector. Note that if we do the identification

[TABLE]

the extended $AdS$ algebra as subalgebra of extended AdS$\oplus$Lorentz, is obtained.  The last five commutations relations describe the coupling of the higher-spin generators $\left\{\mathcal{L}_{ai}^{C_{1}...C_{s}}\text{, }i=0,1\right\}$ to the AdS$\oplus$Lorentz symmetry.

3.3 The extended Maxwell algebra ($\mathcal{B}_{4}^{E}$) as a flat

limit of extended AdS$\oplus$Lorentz ($\mathcal{C}_{4}^{E}$) algebra.

To obtain the extended Maxwell algebra ($\mathcal{B}_{4}^{E}$), we must introduce the $l$ parameter in the definition of the generators in (41), in the form

[TABLE]

Using the extended de Sitter algebra ($\mathcal{C}_{3}^{E}$) (9)-(14), the semigroup (37), and by extending the singular limit $l\longrightarrow\infty$ (flat limit), we obtain by expansion the extension of the Maxwell algebra ($\mathcal{B}_{4}^{E}$) from the resonant and reduced subalgebra (35)

[TABLE]

[TABLE]

Analogously, the first six commutators (50) correspond to the Maxwell algebra and the remaining (51) correspond to the extended sector that describe the coupling of the higher-spin generators $\left\{\mathcal{L}_{ai}^{C_{1}...C_{s}}\text{, }i=0,1\right\}$ to the Maxwell symmetry.

3.4 Extended $\mathcal{C}_{k}^{E}$ algebra as a resonant and reduced

subalgebra of extended de Sitter algebra ($\mathcal{C}_{3}^{E}$).

Following the same methodology; semigroups $S^{(5)}$, $S^{(6)}$, $S^{(7)},...$were obtained, where it was possible to see explicitly that each semigroup was embedded in the next of higher order, in the same way as the semigroup $S^{(2)}$ (24) is embedded in $S^{(3)}$ (30). The semigroups $S^{(5)}$, $S^{(6)}$, $S^{(7)},...$ were obtained using the general multiplication rule shown below in $i$, $ii$, $iii$, $iv$.  The following Lie algebras: $\mathcal{C}_{5}^{E}$ , $\mathcal{C}_{6}^{E}$ , $\mathcal{C}_{7}^{E},...$ and its corresponding flat limits ($l\longrightarrow\infty$) $\mathcal{B}_{5}^{E}$ , $\mathcal{B}_{6}^{E}$ , $\mathcal{B}_{7}^{E},...$ can be obtained directly using the S-expansion method [33]-[36] (appendices A and B) and the corresponding semigroups $S^{(5)}$, $S^{(6)},S^{(7)},...$

Following this inductive method, the generalized extended $\mathcal{C}_{k}^{E}$ algebras can be obtained, using the general semigroup

[TABLE]

with the resonant partition

[TABLE]

and generalizing the multiplication rules used in (24) and (30).

Keeping in mind that $\lambda_{0}$ is the unit element (monoid structure), where

[TABLE]

for all $\lambda_{n}$ $\in S^{(k)}$, we have generalized the multiplication rules in (24) and (30), as follows:

$i)$ Lambda with even index, squared:

[TABLE]

where $n=0,1,2,...$

$ii)$ Lambda with even index by Lambda with even index:

[TABLE]

$iii)$ Lambda with odd index by Lambda with odd index:

[TABLE]

where $n,m=0,1,2,...$

$iv)$ Lambda with even index by Lambda with odd index:

[TABLE]

where $\lambda_{2n}$ , $\lambda_{2n+1}\in S^{(k)}$ and $n=0,1,2,...$

It is straightforward to verify that multiplication rules $i$, $ii$, $iii$ and $iv$ reproduce cases (24), (30) and also  the cases $S^{(5)}$, $S^{(6)}$ and $S^{(7)}.$

Using the partition (55), and applying the S-expansion procedure shown in Appendices A and B (91), [33], [34], we have the resonating subspaces

[TABLE]

and the resonant subalgebra

[TABLE]

Choosing the zero of the semigroup $\lambda_{k-1}=0$, we obtain the reduced resonant subalgebra

[TABLE]

Knowing that, the multiplication rules of $ii$ and $iv$ are compatible with the condition (36), without entering into contradiction we can generalize it, as follows

[TABLE]

where the case $n=1$ and $k=4$ reproduces the condition (36) and the semigroup (30).

We can define the generators as follows

[TABLE]

where $n=0,1,2,..$.

Using the generators (65) and (41), the multiplication rules(57), (58), (59) and (60), the extended $AdS^{E}$ algebra (9)-(14), and the general condition (64); the commutations relations of extended $\mathcal{C}_{k}^{E}$ algebra, are

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Note that when $n=0,1$ and $m=0,1$ the generators of (41) are rescued, where $\mathcal{J}_{0}^{\mu\nu}\rightarrow\mathcal{J}^{\mu\nu}$ and $\mathcal{J}_{2}^{\mu\nu}\rightarrow\mathcal{Z}^{\mu\nu}$ and we obtain the already known commutations relations of extended AdS$\oplus$Lorentz ($\mathcal{C}_{4}^{E}$) algebra (42), (43), where $\mathcal{C}_{4}^{E}\subseteq$ $\mathcal{C}_{k}^{E}$.

3.5 Extended $\mathcal{B}_{k}^{E}$ algebra as a flat limit of

extended $\mathcal{C}_{k}^{E}$ algebra.

The extended $\mathcal{B}_{k}^{E}$ algebras are obtained as the flat limit of the extended $\mathcal{C}_{k}^{E}$ algebras, when $l\longrightarrow\infty.$ In effect, introducing the $l$ parameter in the definition of the generators (65) in the form

[TABLE]

the computing of the commutation relations of the generators (73) and by extending the singular limit $l\longrightarrow\infty$, we obtain the commutation relations of extended and generalized $\mathcal{B}_{k}^{E}$ algebra, which corresponds to the flat limit of the commutation relations of the extended $\mathcal{C}_{k}^{E}$ algebra (66)-(71):

$\circ$ For $n>1$ and $m>0$

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

$\circ$ For $n=0,1$ and $m=0$ is reduced to the already known commutation relations of extended Maxwell algebra ($\mathcal{B}_{4}^{E}\subseteq\mathcal{B}_{k}^{E}$) (50)-(51), where $\mathcal{J}_{0}^{\mu\nu}=\lambda_{0}\mathcal{J}^{\mu\nu}$, $l\mathcal{P}_{1}^{\rho}=\lambda_{1}P^{\mu}$, $l^{2}\mathcal{J}_{2}^{\mu}=\lambda_{1}J^{\mu\nu}$ and $\mathcal{J}_{2}^{\mu\nu}\rightarrow\mathcal{Z}^{\mu\nu}$ of (49).

4 Comments and possible developments

Starting from higher spin extended de Sitter and Conformal Lie algebra ($\mathcal{C}_{3}^{E}$) proposed by [16], [17] we can recover all the other families of higher spin extended $\mathcal{C}_{k}^{E}$ and its flat limit, the $\mathcal{B}_{k}^{E}$ algebras. Specifically, from extended de Sitter algebra ($\mathcal{C}_{3}^{E}$) and using the S-expansion method of Lie algebra (shown in Appendices A and B), we have obtained as a resonant and reduced subalgebras: the extended Poincaré algebra ($\mathcal{B}_{3}^{E}$), the extended AdS$\oplus$Lorentz algebra ($\mathcal{C}_{4}^{E})$ and its flat limit the extended Maxwell algebra ($\mathcal{B}_{4}^{E}$), and their generalizations, the extended $\mathcal{C}_{k}^{E}$ algebra and its corresponding flat limit the extended $\mathcal{B}_{k}^{E}$ algebra [22], [23]. The above results are summarized in the following diagram, i.e,

[TABLE]

[TABLE]

As possible developments we will investigate the irreducible representations of the extended Maxwell algebra ($\mathcal{B}_{4}^{E}$) and extended $\mathcal{B}_{k}^{E}$ family algebras for massless cases. Both the longitudinal and transversal representations would be studied, such as the case of extended (Super)Poincaré algebras in Ref. [16], [17]. Continuous Spin Representations of the Poincaré and Super-Poincaré Groups has been built in Ref. [37], [38].  Another possible developments, we will intend to study the expansions of (Super)symmetric version of extended de Sitter algebra in four space-time dimensions.

5 Acknowledgment

This work was supported by the Research Project, Code DIN 11/2012 of the Universidad Católica de la Santísima Concepción, Chile. R.C. would like to thank to P. Salgado and S. Salgado for his useful comments.

Appendix

Appendix A Brief introduction to the S-expansion method

Given a finite abelian semigroup $S=\left\{\lambda_{0},\lambda_{1},...,\lambda_{N+1}\right\}$, with a commutative and associative composition law $S\times S\rightarrow S$,

[TABLE]

and be the pair $(\mathcal{G};[;])$ a Lie algebra, where $\mathcal{G}$ is a finite dimensional vector space, with basis $\left\{T_{A}\right\}_{A=1}^{\dim G}$ over the field $K$ of the real or complex numbers; and $[;]$ is a rule of composition $\mathcal{G\times G\rightarrow G}$,

[TABLE]

The direct product $\mathcal{B=}S\times\mathcal{G}$ is defined as

[TABLE]

provided with the composition law $[;]_{S}$: $\mathcal{B\times B\rightarrow B}$ defined by

[TABLE]

where

[TABLE]

The prior equation defines the Lie bracket of the S-expanded Lie algebra, where $T_{\left(A,\alpha\right)}=\lambda_{\alpha}T_{A}$ is a base of $\mathcal{B}$, with the composition law (85) and it is called a S-expanded Lie algebra [33]-[36].  The expanded algebra (84) satisfy the linearity, antisymmetry and the Jacobi identity. Indeed, the expanded structure constants satisfy the Jacobi condition, that is to say,

[TABLE]

Appendix B Resonant ($\mathcal{B}_{R}$) and reduced subalgebras ($\mathcal{B}_{R0}$) of $\mathcal{B=}S\times\mathcal{G}$

According to the S-expansion method, a decomposition of the abelian semigroup $S$ can be make, $S=S_{0}\cup S_{1}$ [33], [34], when the algebra $\mathcal{G}$ can be splitting as a direct sum of a subalgebra and symmetric coset, $\mathcal{G=}V_{0}\oplus V_{1}$, that is to say

[TABLE]

and it is said that such decomposition is resonant, when the following conditions are satisfied

[TABLE]

The resonant subalgebras $\mathcal{B}_{R}\subseteq\mathcal{B}$ can be obtained by construction

[TABLE]

where

[TABLE]

and reduced resonant subalgebras, $\mathcal{B}_{R0}\subseteq\mathcal{B}_{R}\subseteq\mathcal{B}$, can be obtained by choosing a zero element of the semigroup, $\lambda_{N+1}:=0_{S}$, and therefore

[TABLE]

This causes some of commutation relations to be null (abelianize) and the process to be equivalent with an Inönü-Wigner contraction.

Therefore, the resonant and reduced subalgebras are not mere copies of the original algebra, but are more complex structures.

For the case when the $\mathcal{G}$ algebra presents a structure of superalgebra; that is to say

[TABLE]

where

[TABLE]

in analogy to the previous case, we can make a resonant decomposition $S=S_{0}\cup S_{1}\cup S_{2}$, which satisfy the following conditions

[TABLE]

Following the S-expansion procedure, we can build the resonant subalgebras

[TABLE]

where

[TABLE]

and the reduced resonant subalgebras $\mathcal{B}_{R0}$, choosing a zero element of the semigroup, e.g. $\lambda_{N+1}:=0_{S}$.