Maxwell kinematical algebras and 3D gravities

Maxw ell kinematical algebras and 3D gra vities P atrick Conc ha ∗ , • , Nelson Gallegos ⋆, • Ev elyn Ro dr ´ ıguez ∗ , • Sebasti´ an Salgado † ∗ Dep artamento de Matem´ atic a y F ´ ısic a Aplic adas, Universidad Cat´ olic a de la Sant ´ ısima Conc ep ci´ on, A lonso de R ib er a 2850, Conc ep ci´ on, Chile. • Grup o de Investigaci´ on en F ´ ısic a T e´ oric a, GIFT, Universidad Cat´ olic a de la Sant ´ ısima Conc ep ci´ on, A lonso de R ib er a 2850, Conc ep ci´ on, Chile. ⋆ Dep artamento de F ´ ısic a, Universidad de Conc ep ci´ on, Casil la 160-C, Conc ep ci´ on, Chile. † Dir e c ci´ on de Investigaci´ on, Universidad Bernar do O’Higgins, Gener al Gana 1702, Santiago, Chile. patrick.concha@ucsc.cl , ngallegos2020@udec.cl erodriguez@ucsc.cl , sebasalg@gmail.com , Abstract In this pap er, we present a Maxwell extension of kinematical Lie algebras by promoting the con traction metho d underlying the Bacry and L´ evy-Leblond cub e to a semigroup expansion framew ork. Within this approach, w e show that b oth non- and ultra-relativistic Maxwell algebras admitting non-degenerate inv arian t bilinear forms can b e systematically obtained from different paren t algebras through a unified expansion scheme, leading to a Maxwellian kinematical cub e. This construction is further generalized to an infinite hierarch y of kinematical algebras. The expansion metho d naturally pro vides the corresponding in v ariant tensors, allowing for the systematic construction of three-dimensional Chern–Simons gravit y theories. Con ten ts 1 In tro duction 1 2 Kinematical Lie algebras 2 3 Maxw ell generalization of kinematical Lie algebras and gravit y actions 5 4 Generalized extended kinematical algebras 15 5 Discussion 18 A Explicit commutation relations of extended kinematical algebras 20 B Explicit commutation relations of the B 5 extended kinematical algebras 21 1 In tro duction Kinematical Lie algebras enco de the transformations b etw een inertial observ ers in spacetime under the assumptions of homogeneity , isotropy , inv ariance under parity and time reversal, and the requiremen t that the transformations relating different inertial frames form non-compact subgroups. These algebras were classified b y Bacry and L´ evy-Leblond, who show ed that they can be organized as the v ertices of a cub e [ 1 ]. Kinematical algebras naturally split in to relativistic and non-Lorentzian symmetry structures. In recen t y ears, non-Lorentzian regimes of (sup er)gravit y theories hav e attracted increasing attention due to their relev ance in a wide range of ph ysical contexts. On the one hand, non-relativistic (Galilean) symmetries play a central role in holograph y [ 2 – 13 ], Ho ˇ ra v a-Lifshitz gravit y [ 14 – 19 ] and effectiv e descriptions of condensed matter systems such as the quan tum Hall effect [ 20 – 24 ]. On the other hand, ultra-relativistic (Carrollian) symmetries ha ve emerged in div erse settings, ranging from tensionless strings and w arp ed CFTs [ 25 – 27 ] to asymptotic symmetries, flat holography , and near-horizon ph ysics [ 28 – 44 ]. A go o d lab oratory to prob e structural asp ects of gra vity in the non-Loren tzian regime is provided b y the three-dimensional mo del formulated in the Chern-Simons (CS) framework, whic h share sev eral features with higher-dimensional gravit y theories, such as black-hole solutions and their asso ciated thermo dynamics [ 45 ]. In this con text, the existence of a non-degenerate in v arian t bilinear form is a crucial requiremen t, as it guaran tees w ell-defined field equations. This condition imp oses non-trivial constrain ts on the underlying symmetry algebra. F or instance, in the non-relativistic case, a consistent CS action requires the extended Bargmann algebra [ 46 , 47 ], which corresp onds to a double cen tral extension of the Galilei algebra. In con trast, in the ultra-relativistic regime, the Carroll algebra already admits a non-degenerate in v ariant tensor [ 48 – 52 ]. More recen tly , a non-v anishing torsion has been incorp orated in to three-dimensional Carrollian gravit y , affecting the non-affinity of null generators and b oundary dynamics [ 53 ]. Matter couplings and conformal approac hes to Carroll gravit y hav e also b een explored in [ 54 – 60 ]. A t the relativistic level, the Maxwell algebra has attracted considerable in terest ov er the past decades, which has b een introduced to describ e a constant electromagnetic field in a Minko wski bac kground [ 61 – 64 ]. The Maxw ell algebra can b e understo o d as an extension and deformation of the P oincar ´ e algebra and has b een extensively studied within the three-dimensional CS gra vity 1 framew ork [ 65 – 69 ]. In particular, recen t analyses of asymptotically flat cosmological solutions in Maxwell CS gravit y hav e shown that the gra vitational Maxw ell field modifies the structure of canonical generators and con tributes non-trivially to the first law of thermo dynamics [ 69 ]. Sup ersymmetric extensions of the Maxw ell algebra, as w ell as higher-spin generalizations, ha ve also been explored [ 70 – 77 ]. In contrast to the P oincar ´ e case, the non-Lorentzian counterpart of the Maxw ell CS gravit y requires a more subtle treatmen t, as b oth the non- and ultra-relativistic regimes generically lead to degeneracies in the inv arian t bilinear form. T o ov ercome this issue, a Maxw ell extension of the extended Bargmann algebra has b een introduced in [ 78 ], obtained as a con traction of the [Maxw ell] ⊕ u (1) 3 algebra. Similarly , in the ultra-relativistic regime, a Maxwellian extended Carroll algebra has b een prop osed in order to ensure a non-degenerate form ulation [ 79 ]. In this work, w e show that the non-degenerate non-Loren tzian Maxwell algebras in tro duced in [ 78 , 79 ] can b e naturally understo od as elements of a Maxwellian generalization of the Bacry and L ´ evy-Leblond cub e, obtained b y replacing the In¨ on ¨ u-Wigner con traction with an expansion pro cedure. In contrast to con tractions, expansion metho ds [ 80 – 83 ] typically increase the n umber of generators of the original algebra. In particular, the semigroup expansion metho d [ 82 ] has pro v en to b e a p ow erful to ol for constructing non-Lorentzian symmetry algebras that admit a non-degenerate in v ariant bilinear form [ 51 ], and therefore for defining consisten t non-Loren tzian (sup er)gra vity theories [ 84 – 92 ]. Remark ably , within this framework it is also p ossible to preserve the dimensionalit y of the original algebra. A particularly illustrativ e example is pro vided by the semigroup S (1) E , whic h repro duces the In¨ on ¨ u–Wigner contraction for an appropriate choice of subspace decomp osition. This observ ation allows one to reinterpret the Bacry and L ´ evy–Leblond cub e 1 as a diagram in which the arro ws correspond to S (1) E expansions rather than con tractions [ 51 ]. It naturally motiv ates the construction of a generalized cub e, where eac h arrow is replaced b y an expansion associated with a higher-order semigroup. The paper is organized as follo ws. In section 2 w e briefly review the kinematical algebras in tro duced in [ 1 ]. Our main results are presented in sections 3 and 4 . In section 3 we construct the Maxw ellian generalization of the kinematical algebras using the semigroup expansion pro cedure and analyze the corresp onding CS gra vit y actions. In section 4 we extend this construction to a broader class of generalized kinematical algebras based on an arbitrary semigroup S ( N ) E . Finnaly , section 5 con tains our conclusions and a discussion of p ossible future directions. 2 Kinematical Lie algebras In this section we briefly review the kinematical Lie algebras classified b y Bacry and L ´ evy- Leblond in [ 1 ]. By construction, kinematical algebras do not describ e the dynamics of particles or their interactions; rather, they specify how reference frames mov e relative to each other in different spacetimes. Nevertheless, spacetime symmetry algebras pla y a cen tral role in gra vitational theories, where they are often promoted to gauge symmetries. A notable example is three-dimensional gravit y , whic h admits a formulation as a Chern-Simons (CS) gauge theory for the AdS group [ 93 – 95 ]. Motiv ated b y this p ersp ective, a systematic study of gauge theories asso ciated with the non- and ultra-relativistic limits of relativistic symmetry algebras b ecomes particularly relev an t for understanding non-Loren tzian regimes of gravit y . In three spacetime dimensions, such theories are naturally describ ed within the CS framew ork, where the fundamental field is a one-form gauge 2 connection A v alued in a Lie algebra g . The action is obtained by integrating the standard CS form, S CS = k 4 π Z  A d A + 2 3 A 3  , (2.1) where ⟨· · · ⟩ denotes an inv ariant, symmetric bilinear form on g . The non-degeneracy of this in v ariant tensor ensures that the equations of motion enforce the v anishing of the gauge curv ature F = d A + A 2 . As a consequence, the existence of a non-degenerate in v arian t bilinear form b ecomes a crucial ingredient in the construction of CS gravit y theories with w ell-defined field equations, and m ust therefore b e analyzed on a case-by-case basis for each algebra under consideration [ 48 , 51 ]. Let us consider the three-dimensional AdS algebra so (2 , 2), whic h serves as the starting p oint for the construction of the Bacry and L´ evy-Leblond cub e. This algebra is spanned by the generators { J A , P A } , where the indices A, B = 0 , 1 , 2 are raised and low ered with the Minko wski metric η AB = diag( − , + , +). These generators satisfy the commutation relations: h ˜ J A , ˜ J B i = ϵ AB C ˜ J C , h ˜ J A , ˜ P B i = ϵ AB C ˜ P C , h ˜ P A , ˜ P B i = ϵ AB C ˜ J C , (2.2) where the Loren tzian Levi-Civita pseudotensor is defined b y ϵ 012 = 1. The P oincar´ e algebra is obtained from ( 2.2 ) by means of the In¨ on ¨ u-Wigner contraction, implemen ted through the rescaling ˜ P A → ℓ ˜ P A follo wed by the limit ℓ → ∞ . This pro cedure corresp onds to the v anishing cosmological constan t limit, where the contraction parameter is related to the cosmological constan t as ℓ − 2 = − Λ. In this flat limit, the translational sector b ecomes ab elian, namely h ˜ P A , ˜ P B i = 0. Both the AdS and Poincar ´ e algebras are Lorentzian kinematical algebras, describing transformations b etw een relativistic observ ers in spacetimes of constant negative and v anishing curv ature, resp ectiv ely . Non-Loren tzian kinematical algebras arise from the AdS and Poincar ´ e algebras by considering the ph ysical limits in whic h the sp eed of ligh t is taken to zero or to infinity . Implementing these non-Loren tzian limits requires splitting the Lorentz-co v ariant index in to temp oral and spatial comp onen ts, A → (0 , a ). Accordingly , the generators are decomp osed as ˜ J A =  ˜ J 0 , ˜ J a  ≡ ( J, G a ) , ˜ P A =  ˜ P 0 , ˜ P a  ≡ ( H, P a ) , (2.3) where the spatial index tak es the v alues a = 1 , 2. Here, J generates spatial rotations in t wo- dimensional Euclidean space, while G a generates b o ost transformations relating space and time for differen t observ ers. The generator H corresp onds to time translations, and P a generates spatial translations. In terms of these generators, the AdS and Poincar ´ e algebras can b e rewritten as follo ws: AdS : [ J, G a ] = ϵ ab G b , [ H , G a ] = ϵ ab P b , [ G a , G b ] = − ϵ ab J , [ H , P a ] = ϵ ab G b , [ J, P a ] = ϵ ab P b , [ P a , P b ] = − ϵ ab J , [ G a , P b ] = − ϵ ab H , (2.4) P oincar´ e : [ J, G a ] = ϵ ab G b , [ H , G a ] = ϵ ab P b , [ G a , G b ] = − ϵ ab J , [ J, P a ] = ϵ ab P b , [ G a , P b ] = − ϵ ab H , (2.5) 3 where ϵ ab ≡ ϵ 0 ab , ϵ ab ≡ ϵ 0 ab . The non- and ultra-relativistic limits are in tro duced through sp eed–space and sp eed–time con tractions, resp ectively . These limits are defined by the following rescalings: Sp eed-space rescaling: P a → σ P a , G a → σ G a , Sp eed-time rescaling: G a → κ G a , H → κ H . (2.6) The scaling parameters are related to the sp eed of ligh t as σ = c and κ = c − 1 . Applying b oth con tractions to the AdS and Poincar ´ e algebras gives rise to four distinct kinematical Lie algebras. The non-relativistic versions of these algebras are obtained b y considering the sp eed-space rescaling in the limit σ → ∞ . They are given by [ 1 ]: Newton-Ho ok e : [ J, G a ] = ϵ ab G b , [ H, G a ] = ϵ ab P b , [ J, P a ] = ϵ ab P b , [ H , P a ] = ϵ ab G b , (2.7) Galilei : [ J , G a ] = ϵ ab G b , [ H, G a ] = ϵ ab P b , [ J, P a ] = ϵ ab P b . (2.8) On the other hand, the Carrollian limits of the AdS and P oincar´ e algebras are obtained by applying the sp eed-time rescaling and taking the limit κ → ∞ . The resulting algebras are giv en by [ 1 ]: P ara-Poincar ´ e : [ J, G b ] = ϵ ab G b , [ G a , P a ] = − ϵ ab H , [ J, P a ] = ϵ ab P b , [ P a , P b ] = − ϵ ab J , [ H , P a ] = ϵ ab G b , (2.9) Carroll : [ J, G b ] = ϵ ab G b , [ G a , P a ] = − ϵ ab H , [ J, P a ] = ϵ ab P b . (2.10) Moreo ver, the P ara-Poincar ´ e (also known as the AdS-Carroll) and Carroll algebras are related through the v anishing cosmological constant limit. The cosmological constan t can b e reintroduced b y means of the rescaling ( H , P a ) → ( ℓH , ℓP a ) in ( 2.9 ) , follo wed by the limit ℓ → ∞ , whic h leads to the Carroll algebra. An analogous reasoning can b e applied to the Newton-Hooke and Para-Galilei algebras, leading to the Galilei and static algebras, resp ectiv ely . Figure 1: Bacry and L´ evy-Leblond cub e of kinematical algebras [ 1 ]. 4 The complete set of contractions is schematically illustrated by the Bacry and L´ evy-Leblond cub e [ 1 ] of kinematical algebras (see Fig. 1 ). Let us note that the cub e also includes the so- called static and para-Galilean algebras, whic h can b e obtained as sp eed-space con tractions of the Carroll and AdS–Carroll algebras (or, equiv alen tly , as sp eed-time contractions of the Galilean and Newton-Ho ok e algebras). 3 Maxw ell generalization of kinematical Lie algebras and gra vit y actions The non-Loren tzian limits of the Maxw ell algebra ha ve already b een in v estigated in [ 78 , 79 ], where it w as shown that additional cen tral c harges must b e introduced in b oth the non- and ultra-relativistic regimes in order to av oid degeneracies in the inv ariant bilinear form. In this section, w e show that b oth non-degenerate algebras can b e embedded into a Maxwellian version of the Bacry and L´ evy-Leblond cub e [ 1 ], in whic h the standard con traction scheme is replaced by a more general algebraic construction. With the aim of constructing a Maxw ellian version of the kinematical Lie algebras [ 1 ] without degeneracy , we b egin by considering the general structure of an S (2) E -expansion, whic h replaces the usual contraction limit. W e introduce a general Lie algebra g = span { T A } = V 0 ⊕ V 1 endo wed with a subspace decomp osition [ V 0 , V 0 ] ⊂ V 0 , [ V 0 , V 1 ] ⊂ V 1 , [ V 1 , V 1 ] ⊂ V 0 . (3.1) W e further consider the ab elian semigroup S (2) E = { λ 0 , λ 1 , λ 2 , λ 3 } , (3.2) endo wed with the following multiplication law λ α λ β =  λ α + β if α + β ≤ 3 , λ 3 if α + β > 3 . (3.3) It is straightforw ard to verify that the decomp osition ( 3.1 ) is in resonance with the splitting of the semigroup S (2) E in to the subsets S 0 = { λ 0 , λ 2 , λ 3 } , S 1 = { λ 1 , λ 3 } , (3.4) whic h satisfy S 0 · S 0 ⊂ S 0 , S 0 · S 1 ⊂ S 1 , S 1 · S 1 ⊂ S 0 . (3.5) This resonance condition allows for the construction of the corresp onding resonant expanded subalgebra G = ( S 0 × V 0 ) ⊕ ( S 1 × V 1 ) . (3.6) Moreo ver, since the semigroup admits a zero element λ 3 , satisfying λ 3 λ i = λ 3 for an y λ i ∈ S (2) E , one can consisten tly p erform a 0 S -reduction b y imp osing the condition λ 3 × T A = 0. The resulting algebra will b e refer as a resonan t 0 S -reduced S (2) E -expanded algebra. 5 Our starting p oint is the AdS algebra and its non-Lorentzian counterparts. As shown in [ 51 , 92 ], the S (2) E -expansion applied to the AdS algebra, upon considering appropriate speed-space and sp eed-time subspace decomp ositions, reproduces a family of extended kinematical algebras (see Fig. 2 ). At the non-relativistic lev el, the resulting algebra corresp onds to the extended Newton- Ho ok e algebra [ 96 – 102 ], whic h, in the v anishing cosmological constant limit Λ → 0, reduces to the extended Bargmann algebra [ 46 , 47 ]. In the ultra-relativistic regime, the expansion yields to the extended P ara-P oincar´ e algebra (or extended AdS-Carroll algebra 1 ) [ 51 ], whose flat limit giv es rise to an extended Carroll algebra. On the other hand, tw o successive expansions lead to an extended AdS-Static algebra 2 . Eac h of these extended kinematical algebras, whose explicit commutation relations are listed in App endix A , admits a non-degenerate in v ariant tensor, which in turn allows for the construction of a w ell-defined gra vitational CS action. Figure 2: Extended kinematical algebras starting from the AdS algebra [ 51 ]. The Maxw ellian generalization can then b e obtained by promoting the flat limit Λ → 0 to an expansion pro cedure (see Fig. 3 ). In particular, the Maxwell kinematical algebras arise from the application of a resonan t S (2) E -expansion follo wed b y a 0 S -reduction. Remark ably , as illustrated b y the cub e in Fig. 3 , the non-Lorentzian Maxwellian algebras can also b e recov ered from the relativistic Maxwell algebra, inheriting the expansion relations already presen t in the corresp onding extended kinematical algebras. Figure 3: Maxwellian generalization of the extended kinematical algebras. 1 Lab eled as eAdS-C in Fig. 2 . 2 Also denoted as general Para-Bargmann in [ 51 ] and lab eled as eAdS-S in Fig. 2 . 6 Maxw ell algebra Let us start with a subspace decomp osition of the AdS algebra ( 2.4 ) defined b y V 0 = { J, G a } , V 1 = { H , P a } , (3.7) whic h satisfies a Z 2 gradation of the form ( 3.1 ) . The Maxwell algebra is then obtained b y p erforming a resonan t S (2) E -expansion of the AdS algebra, follo wed by the application of the 0 S -reduction. The Maxw ell generators are related to those of so (2 , 2) through the semigroup elements, as in T able 1 . λ 3 λ 2 Z , Z a λ 1 H , P a λ 0 J , G a J , G a H , P a T able 1: Maxwell generators in terms of the AdS ones and the semigroup elemen ts. The comm utation relations of the Maxwell algebra are obtained by combining the so (2 , 2) comm utators and the multiplication law of the semigroup S (2) E ( 3.3 ), [ J , G a ] = ϵ ab G b , [ H , G a ] = ϵ ab P b , [ G a , G b ] = − ϵ ab J , [ J , P a ] = ϵ ab P b , [ H , P a ] = ϵ ab Z b , [ G a , P b ] = − ϵ ab H , [ J , Z a ] = ϵ ab Z b , [ Z , G a ] = ϵ ab Z b , [ P a , P b ] = − ϵ ab Z , [ G a , Z b ] = − ϵ ab Z . (3.8) It is worth noting that the Maxw ell algebra ( 3.8 ) is isomorphic to the extended AdS-Carroll algebra (see T able 13 ), an isomorphism that b ecomes manifest up on the following identification of generators 3 : G a ↔ P a , P a ↔ G a , Z ↔ C , Z a ↔ T a . (3.9) A CS gra vity action based on the Maxw ell algebra can then b e written in terms of a gauge connection one-form, A = W J + V H + W a G a + V a P a + K Z + K a Z a , (3.10) where W is the spin-connection for b o osts, W a represen ts the spatial spin-connection, V corresp onds to the time-like vielb ein, V a is the spatial vielb ein, K represen ts the time-like gravitational Maxw ell field and K a is the spatial gra vitational Maxwell field. The Maxwell algebra admits a non-degenerate in v ariant tensor whose non-v anishing comp onents read ⟨ JJ ⟩ = − α 0 , ⟨ G a G b ⟩ = α 0 δ ab , ⟨ JH ⟩ = − α 1 , ⟨ JZ ⟩ = − α 2 , ⟨ G a P b ⟩ = α 1 δ ab , ⟨ HH ⟩ = − α 2 , ⟨ G a Z b ⟩ = α 2 δ ab , ⟨ P a P b ⟩ = α 2 δ ab , (3.11) 3 This isomorphism motiv ates the denomination Par a-Maxwel l algebra for the extended AdS-Carroll algebra. 7 where the non-degeneracy condition requires α 2  = 0. The CS action for the Maxw ell algebra ( 3.8 ) is then obtained by substituting the one-form gauge connection ( 3.10 ) and the non-v anishing comp onen ts of the inv ariant tensor ( 3.11 ) into the general CS expression ( 2.1 ): I Max = k 4 π Z α 0  − W dW + W a R a  W b  + α 1  2 V a R a  W b  − 2 V R ( W )  + α 2  2 K a R a  W b  + V a R a  V b  − V R ( V ) − 2 K R ( W )  . (3.12) Here, k = 1 / 4 G is the level of the theory related to the gravitational constant G and R ( W ) = dW + 1 2 ϵ ab W a W b , R a  W b  = dW a + ϵ ab W W b , R ( V ) = dV + ϵ ab W a V b , R a  V b  = dV a + ϵ ab W V b + ϵ ab V W b . (3.13) The non-degeneracy ensures that the field equations are giv en b y the v anishing of the curv ature t wo-forms ( 3.13 ) as well as those related to the gravitational Maxwell field, R ( K ) = dK + 1 2 ϵ ab W a K b + 1 2 ϵ ab V a V b , R a  K b  = dK a + ϵ ab W K b + ϵ ab V V b + ϵ ab K W b . (3.14) While the v anishing of the curv atures ( 3.13 ) describ es a locally flat Riemannian geometry , the conditions imp osed by the v anishing of the curv atures ( 3.14 ) lead to non-trivial effects when compared to General Relativity , which hav e b een extensively studied in [ 65 – 69 ]. A cosmological constan t can b e included in the Maxwell CS gra vit y theory but requires a deformation of the algebra to the so-called AdS-Loren tz algebra [ 103 – 105 ]. As a final remark, it is worth noticing that the CS action ( 3.12 ) can b e reco vered from the so (2 , 2) CS action, I AdS = k 4 π Z µ 0  − W dW + W a R a  W b  + V a R a  V b  − V R ( V )  + µ 1  2 V a R a  W b  + ϵ ab V V a V b − 2 V R ( W )  . (3.15) Indeed, the expansion pro cedure can b e made by identifying the Maxw ell gauge fields in terms of the so (2 , 2) ones through the semigroup elemen ts as W = λ 0 W , W a = λ 0 W a , V = λ 1 V , V a = λ 1 V a , K = λ 2 W , K a = λ 2 W a , (3.16) and expressing the α ’s constan t in terms of the so (2 , 2) ones as α 0 = λ 0 µ 0 , α 1 = λ 1 µ 1 , α 2 = λ 2 µ 0 . (3.17) 8 Maxw ellian extended Bargmann algebra Here we show that the Maxwellian version of the extended Bargmann (MEB) algebra [ 78 ] admits t wo distinct origins within the S -expansion framework. In particular, it can b e obtained either from the extended Newton-Ho oke algebra (see T able 12 ) or, alternatively , from the Maxw ell algebra ( 3.8 ) , as summarized in Fig. 3 . As in the previous case, the S-expansion requires a decomp osition in to subspaces (see T able 2 ) satisfying a Z 2 -graded Lie algebra structure of the form ( 3.1 ). Subspaces Extended Newton-Ho ok e origin Maxw ell origin V 0 J, G a , S J , H , Z V 1 H , P a , M G a , P a , Z a T able 2: Subspaces decomp osition of the extended Newton-Ho oke and Maxwell algebra. The commutation relations of the MEB algebra are then obtained by considering a resonant S (2) E -expansion, follow ed by the corresp onding 0 S -reduction. In particular, the MEB generators are related to those of the extended Newton-Ho oke or Maxwell algebras through the semigroup elemen ts, as shown in T able 3 . Then, the commutators of the MEB algebra can b e obtained by Extended Newton-Ho ok e origin Maxw ell origin λ 3 λ 2 Z , Z a , T S , M , T λ 1 H , P a , M G a , P a , Z a λ 0 J , G a , S J , H , Z J , G a , S H , P a , M J , H , Z G a , P a , Z a T able 3: MEB generators expressed in terms of the generators of the extended Newton-Ho oke and Maxw ell algebras through the S (2) E semigroup elemen ts. com bining either the extended Newton-Ho oke or the Maxwell comm utators with the multiplication la w of the semigroup S (2) E ( 3.3 ), [ J , G a ] = ϵ ab G b , [ H , G a ] = ϵ ab P b , [ G a , G b ] = − ϵ ab S , [ J , P a ] = ϵ ab P b , [ H , P a ] = ϵ ab Z b , [ G a , P b ] = − ϵ ab M , [ J , Z a ] = ϵ ab Z b , [ Z , G a ] = ϵ ab Z b , [ P a , P b ] = − ϵ ab T , [ G a , Z b ] = − ϵ ab T . (3.18) This construction sho ws that the MEB algebra can b e understo o d either as a Maxwell-t yp e extension of the extended Newton-Ho oke algebra or, alternativ ely , as a non-relativistic expansion of the Maxw ell algebra itself 4 , as summarized in T able 2 . The presence of the additional generators { Z , Z a , T } reflects the non-trivial commutator structure induced by the semigroup. The MEB algebra, unlike the extended Bargmann one (see T able 12 ), cannot b e recov ered as an extension of the Galilei algebra 4 The present construction corresp onds to a finite sub case of the infinite-dimensional Galilean expansion of the Maxw ell algebra discussed in [ 106 ]. 9 ( 2.8 ) . Interestingly , the MEB algebra ( 3.18 ) is isomorphic to the extended AdS-Static algebra (see T able 13 ) up on the following identification of the generators: G a ↔ P a , P a ↔ G a , Z ↔ C , T ↔ B , Z a ↔ T a . (3.19) This observ ation suggests relab eling the extended AdS-Static algebra as the Par a-MEB algebra. The MEB algebra admits the follo wing non-v anishing components of the inv ariant tensor: ⟨ JS ⟩ = − β 0 , ⟨ G a G b ⟩ = β 0 δ ab , ⟨ JM ⟩ = − β 1 , ⟨ JT ⟩ = − β 2 , ⟨ G a P b ⟩ = β 1 δ ab , ⟨ HS ⟩ = − β 1 , ⟨ SZ ⟩ = − β 2 ⟨ P a P b ⟩ = β 2 δ ab , ⟨ HM ⟩ = − β 2 , ⟨ G a Z b ⟩ = β 2 δ ab , (3.20) where the non-degeneracy condition of the in v arian t tensor requires β 2  = 0. The gauge connection one-form for the MEB algebra reads A = ω J + ω a G a + τ H + e a P a + k Z + k a Z a + s S + m M + t T . (3.21) The CS gra vity action for the Maxwell algebra [ 78 ] is obtained b y inserting the gauge connection one-form ( 3.21 ) and the in v ariant tensor ( 3.20 ) into the general CS expression ( 2.1 ), I MEB = k 4 π Z β 0  − sR ( ω ) + ω a R a  ω b  + β 1  2 e a R a  ω b  − 2 mR ( ω ) − 2 τ R ( s )  + β 2  k a R a  ω b  + ω a R a  k b  + e a R a  e b  − 2 sR ( k ) − 2 mR ( τ ) − 2 tR ( ω )  , (3.22) where R ( ω ) = dω , R a  ω b  = dω a + ϵ ab ω ω b , R ( τ ) = dτ , R a  e b  = de a + ϵ ab ω e b + ϵ ab τ ω b , R ( k ) = dk , R a  k b  = dk a + ϵ ab ω k b + ϵ ab τ e b + ϵ ab k ω b , R ( s ) = ds + 1 2 ϵ ab ω a ω b . (3.23) It is worth emphasizing that the first tw o sectors, prop ortional to β 0 and β 1 , reproduce the most general CS action for the extended Bargmann algebra [ 88 , 107 ]. The Maxw ellian con tribution app ears explicitly in the β 2 sector. As its relativistic v ersion, a cosmological constant can be included deforming the symmetry algebra to an enlarged extended Bargmann algebra [ 86 ]. The field equations, for β 2  = 0, are given by the v anishing of the MEB curv ature tw o-forms represented b y ( 3.23 ) and R ( m ) = dm + ϵ ab ω a e b , R ( t ) = dt + ϵ ab ω a k b + 1 2 ϵ ab e a e b . (3.24) The CS action ( 3.22 ) for the MEB algebra can alternativ ely b e derived from the most general extended Newton-Ho ok e CS action [ 88 , 107 ], I eNH = k 4 π Z ν 0  − sR ( ω ) + ω a R a  ω b  + e a R a  e b  − 2 mR ( τ )  10 + ν 1  2 e a R a  ω b  − 2 mR ( ω ) − 2 τ R ( s ) + ϵ ab τ e a e b  , (3.25) b y expressing the MEB gauge fields in terms of the extended Newton-Ho ok e ones and the semigroup elemen ts as ω = λ 0 ω , ω a = λ 0 ω a , s = λ 0 s , τ = λ 1 τ , e a = λ 1 e a , m = λ 1 m , k = λ 2 ω , k a = λ 2 ω a , t = λ 2 s , (3.26) and iden tifying the β ’s constan t as β 0 = λ 0 ν 0 , β 1 = λ 1 ν 1 , β 2 = λ 2 ν 0 . (3.27) Similarly , the MEB CS action ( 3.22 ) can b e obtained from the Maxwell CS gra vity action ( 3.12 ) b y expressing the MEB gauge fields in terms of the Maxw ell ones through the S (2) E semigroup elements, according to the iden tifications of their resp ective generators summarized in T able 3 . Maxw ellian extended Carroll algebra A t the ultra-relativistic level, we show that the non-degenerate Maxwellian extension of the Carroll algebra, denoted as the Maxw ellian extended Carroll (MEC) algebra [ 79 ], can b e obtained either from the extended AdS-Carroll algebra (see T able 13 ) or from the Maxw ell algebra ( 3.8 ) , as illustrated in Fig. 3 . The MEC algebra is obtained by p erforming a resonant S (2) E -expansion of the starting algebra follo wed by a 0 S -reduction. T o this end, w e first consider the subspace decomp osition sho wn in T able 4 . Subspaces Extended AdS-Carroll origin Maxwell origin V 0 J, G a , C J , P a , Z V 1 H , P a , T a G a , H , Z a T able 4: Subspaces decomp osition of the extended AdS-Carroll and Maxwell algebra. The MEC generators can b e reco vered from the extended AdS-Carroll or the Maxwell generators through the semigroup elemen ts, as displa yed in T able 5 . Extended AdS-Carroll origin Maxwell origin λ 3 λ 2 Z , Z a , L C , T a , L λ 1 H , P a , T a G a , H , Z a λ 0 J , G a , C J , P a , Z J , G a , C H , P a , T a J , P a , Z G a , H , Z a T able 5: MEC generators expressed in terms of the extended AdS-Carroll and Maxwell generators through the S (2) E semigroup elemen ts. 11 The MEC algebra satisfies the follo wing comm utation relations: [ J , G a ] = ϵ ab G b , [ Z , G a ] = ϵ ab Z b , [ P a , P b ] = − ϵ ab Z , [ J , P a ] = ϵ ab P b , [ H , P a ] = ϵ ab Z b , [ G a , P b ] = − ϵ ab H , [ J , Z a ] = ϵ ab Z b , [ H , G a ] = ϵ ab T b , [ G a , G b ] = − ϵ ab C , [ J , T a ] = ϵ ab T b , [ C , P a ] = ϵ ab T b , [ G a , Z b ] = − ϵ ab L , [ P a , T b ] = − ϵ ab L . (3.28) These comm utation relations are obtained by combining either the extended AdS-Carroll or the Maxw ell comm utators with the m ultiplication law of the semigroup S (2) E giv en in ( 3.3 ) . Let us note that the MEC algebra, which contains 13 generators, is not isomorphic to the MEB algebra ( 3.18 ) , whic h is spanned b y 12 generators. In particular, the MEC algebra is characterized by the presence of a central charge L , whose presence ensures the non-degenercy of the inv ariant tensor, whic h reads ⟨ JJ ⟩ = − γ 0 , ⟨ G a G b ⟩ = γ 0 δ ab , ⟨ JC ⟩ = − γ 0 , ⟨ JH ⟩ = − γ 1 , ⟨ G a P b ⟩ = γ 1 δ ab , ⟨ JZ ⟩ = − γ 2 , ⟨ JL ⟩ = − γ 2 , ⟨ P a P b ⟩ = γ 2 δ ab , ⟨ HH ⟩ = − γ 2 , ⟨ CZ ⟩ = − γ 2 , ⟨ G a Z b ⟩ = γ 2 δ ab , ⟨ P a T b ⟩ = γ 2 δ ab . (3.29) The MEC CS gra vity action is written in terms of the gauge connection one-form, A = ω J + ω a G a + τ H + e a P a + k Z + k a Z a + ς C + t a T a + l L . (3.30) The MEC CS action, first in tro duced in [ 79 ], reads I MEC = k 4 π Z γ 0  − ω R ( ω ) + ω a R a  ω b  − 2 ς R ( ω )  + γ 1  2 e a R a  ω b  − 2 τ R ( ω )  + γ 2  k a R a  ω b  + ω a R a  k b  + e a R a  e b  − 2 ς R ( k ) − 2 l R ( ω ) − 2 k R ( ω ) +2 t a R  e b  − τ R ( τ )  , (3.31) where R ( ω ) = dω , R a  ω b  = dω a + ϵ ab ω ω b , R ( τ ) = dτ + ϵ ab ω a e b , R a  e b  = de a + ϵ ab ω e b , R ( k ) = dk + 1 2 ϵ ab e a e b , R a  k b  = dk a + ϵ ab ω k b + ϵ ab τ e b + ϵ ab k ω b . (3.32) The non-degeneracy of the inv ariant tensor ( 3.29 ) ensures that the field equations are given by the v anishing of the MEC curv ature tw o-forms represented by ( 3.32 ) and R ( ς ) = dς + 1 2 ϵ ab ω a ω b , R ( l ) = dl + ϵ ab ω a k b + ϵ ab e a t b , R a  t b  = dt a + ϵ ab ω t b + ϵ ab ς e b + ϵ ab τ ω b . (3.33) 12 As in the original Carroll algebra, the equations of motion are c haracterized b y a non-v anishing temp oral torsion, dτ = − ϵ ab ω a e b . (3.34) Nonetheless, the Maxw ellian extension introduces additional gauge fields { ς , t a , l } , asso ciated with the generators { C , T a , L } , which are required in order to ensure a non-degenerate inv ariant tensor and thus a well-defined set of field equations. While the role of Maxwell gauge fields at the relativistic level has been explored in differen t con texts [ 66 , 67 , 69 , 77 ], the physical in terpretation of the additional con tent in the Carrollian regime remains largely op en. It is worth p ointing out that the MEC CS action ( 3.31 ) can b e directly obtained from the extended AdS-Carroll CS action I eAdS-C = k 4 π Z σ 0  e a R a  e b  − ω R ( ω )  + σ 1  2 e a R a  ω b  − 2 τ R ( ω ) + ϵ ab τ e a e b  + σ 2  ω a R a  ω b  − 2 τ R ( τ ) − 2 ς R ( ω ) + e a R a  t b  + t a R a  e b  , (3.35) up on implemen ting the field identifications induced by the S (2) E semigroup elemen ts, ω = λ 0 ω , ω a = λ 0 ω a , ς = λ 0 ς , τ = λ 1 τ , e a = λ 0 e a , t a = λ 1 t a , k = λ 2 ω , k a = λ 2 ω a , l = λ 2 ς , (3.36) together with the iden tification of the coupling constants, γ 0 = λ 0 σ 0 , γ 1 = λ 1 σ 1 , γ 2 = λ 2 σ 2 . (3.37) In a similar manner, the MEC action ( 3.31 ) can also b e derived from the Maxwell CS gravit y action ( 3.12 ) b y expressing the MEC gauge fields in terms of the Maxwell ones through the S (2) E semigroup elemen ts, according to the generator identifications display ed in T able 5 . Maxw ellian extended static algebra F or completeness, w e close this section by in tro ducing a Maxw ellian v ersion of the extended static algebra, which we denote as the MES algebra. Although this symmetry algebra has not b een previously discussed in the literature, w e show that the S -expansion framework provides three indep enden t constructions leading to its commutation relations. In particular, the MES algebra can b e obtained from the extended AdS-static, the MEB and the MEC algebras (see T able 6 ). Subspaces Extended AdS-static MEB MEC origin origin origin V 0 J, G a , C, S, B J , P a , Z , S , T J , C , Z , L , H V 1 H , P a , T a , M G a , H , Z a , M G a , P a , T a , Z a T able 6: Subspaces decomp osition of the extended AdS-static, MEB and MEC algebra. 13 Extended AdS-static origin MEB origin MEC origin λ 3 λ 2 Z , Z a , L , T , Y C , T a , L , B , Y S , B , T , Y , M λ 1 H , P a , T a , M G a , H , Z a , M G a , P a , T a , Z a λ 0 J , G a , C , S , B J , P a , Z , S , T J , C , Z , L , H J, G a , C, S, B H , P a , T a , M J , P a , Z , S , T G a , H , Z a , M J , C , Z , L , H G a , P a , T a , Z a T able 7: MES generators expressed in terms of the extended AdS-static, MEB and MEC generators through the S (2) E semigroup elemen ts. The MES algebra arises from a resonant S (2) E -expansion of any of the starting algebras, follo wed b y a 0 S -reduction. The resulting MES generators are related to the generators of the parent algebras through the semigroup elemen ts, as summarized in T able 7 . The comm utation relations of the MES algebra follow from combining the comm utators of the c hosen starting algebra with the m ultiplication law of the S (2) E semigroup ( 3.3 ): [ J , G a ] = ϵ ab G b , [ Z , G a ] = ϵ ab Z b , [ P a , P b ] = − ϵ ab T , [ J , P a ] = ϵ ab P b , [ H , P a ] = ϵ ab Z b , [ G a , P b ] = − ϵ ab M , [ J , Z a ] = ϵ ab Z b , [ H , G a ] = ϵ ab T b , [ G a , G b ] = − ϵ ab B , [ J , T a ] = ϵ ab T b , [ C , P a ] = ϵ ab T b , [ G a , Z b ] = − ϵ ab Y , [ P a , T b ] = − ϵ ab Y . (3.38) Let us note that the MES algebra con tains tw o additional u (1) central generators, S and L , satisfying [ X , S ] = [ X , L ] = 0 , ∀ X ∈ mes . (3.39) Although S and L do not en ter in the non-trivial commutators in ( 3.38 ) , their presence is essential to ensure the non-degeneracy of the in v arian t bilinear form. In particular, the MES algebra admits the follo wing non-v anishing comp onents of the inv ariant tensor: ⟨ JS ⟩ = − ζ 0 , ⟨ G a G b ⟩ = ζ 0 δ ab , ⟨ JB ⟩ = − ζ 0 , ⟨ SC ⟩ = − ζ 0 , ⟨ G a P b ⟩ = ζ 1 δ ab , ⟨ JM ⟩ = − ζ 1 , ⟨ SH ⟩ = − ζ 1 , ⟨ P a P b ⟩ = ζ 2 δ ab , ⟨ JT ⟩ = − ζ 2 , ⟨ SZ ⟩ = − ζ 2 , ⟨ G a Z b ⟩ = ζ 2 δ ab , ⟨ JY ⟩ = − ζ 2 , ⟨ SL ⟩ = − ζ 2 , ⟨ P a T b ⟩ = ζ 2 δ ab , ⟨ HM ⟩ = − ζ 2 , ⟨ CT ⟩ = − ζ 2 , ⟨ BZ ⟩ = − ζ 2 . (3.40) where non-degeneracy requires ζ 2  = 0. Although a Maxwellian static algebra without the generators { C , L , B , Y , T a } still satisfies the Jacobi iden tity , it do es not admit a non-degenerate inv ariant bilinear form. Therefore, the minimal con tent required to obtain a consistent Maxwellian extension of the static algebra, and consequently a w ell-defined CS action, is precisely the complete set of generators in tro duced here. The corresp onding CS action can b e constructed straightforw ardly from ( 3.40 ) and the gauge connection asso ciated with the MES algebra. Giv en its lengthy structure, w e refrain from 14 presen ting it explicitly . Moreov er, it may b e deriv ed directly from the extended AdS-static, MEB or MEC CS gra vity actions by expressing the MES gauge field in terms of the corresp onding parent algebra through the S (2) E semigroup elemen ts, according to the generator identifications display ed in T able 7 . 4 Generalized extended kinematical algebras In this section, w e sho w that the original cub e in tro duced b y Bacry and L ´ evy-Leblond [ 1 ], together with its Maxw ellian extension discussed in the previous section, admits a natural generalization through the use of an arbitrary semigroup S ( N ) E . This construction gives rise to an infinite hierarc hy of generalized kinematical cub es asso ciated with the non-Loren tzian sector of the so-called B k algebras [ 108 – 110 ]. The B k generalization of the extended kinematical Lie algebras is schematically illustrated in Fig. 4 , where the v anishing cosmological constan t limit of the the cub e of Bacry and L ´ evy-Leblond [ 1 ] is extended by means of an S ( N ) E -expansion with N = k − 2. In this construction, the S (2) E -expansion is preserved along b oth the non-relativistic and ultra-relativistic directions in order to obtain a non-degenerate non-Lorentzian coun terpart of the B k algebra. Higher-order semigroups along the non-relativistic or ultra-relativistic directions of the cub e sho wn in Fig. 4 w ould instead generate p ost-Newtonian or post-Carrollian extensions of the B k algebra. Suc h generalizations, ho wev er, lie b eyond the scop e of the present work. Figure 4: B k generalization of the extended kinematical algebras. Our starting p oin t consists of the extended kinematical algebras summarized in Fig. 2 . W e then consider, for each of these algebras, a subspace decomp osition satisfying a Z 2 -graded Lie algebra structure, as displa yed in T able 8 . Subspaces AdS origin eNH origin eAdS-C origin eAdS-S origin V 0 J, G a J, G a , S J, G a , C J, G a , C, S, B V 1 H , P a H , P a , M H , P a , T a H , P a , T a , M T able 8: Subspace decomp ositions of the extended kinematical algebras. Let us no w consider the semigroup S ( N ) E = { λ 0 , λ 1 , · · · , λ N +1 } , whic h ob eys the multiplication 15 la w λ α λ β = ( λ α + β if α + β ≤ N + 1 , λ N +1 if α + β > N + 1 . (4.1) A subset decomp osition of the semigroup S ( N ) E , whic h is said to be resonan t with the subspace splitting giv en in T able 8 , is defined by S 0 = { λ 2 i } ∪ { λ N +1 } , S 1 = { λ 2 i +1 } ∪ { λ N +1 } , (4.2) with i = 0 , 1 , 2 , . . . , [ N / 2], where [ · ] denotes the integer part. The resonant condition then allows for the construction of a resonant S ( N ) E -expanded subalgebra given by ( 3.6 ) . The B k algebra and its corresp onding non-Loren tzian counterparts can b e obtained from the extended kinematical algebras (see Fig. 2 ) by applying a resonant S ( N ) E -expansion follow ed b y the corresp onding 0 S -reduction. The expanded generators are related to the generators of the extended kinematical algebras through the semigroup elemen ts, as summarized in T able 9 . Expanded Generators AdS origin eNH origin eAdS-C origin eAdS-S origin J ( m ) λ 2 m J λ 2 m J λ 2 m J λ 2 m J G ( m ) a λ 2 m G a λ 2 m G a λ 2 m G a λ 2 m G a S ( m ) - λ 2 m S - λ 2 m S C ( m ) - - λ 2 m C λ 2 m C B ( m ) - - - λ 2 m B H ( m ) λ 2 m +1 H λ 2 m +1 H λ 2 m +1 H λ 2 m +1 H P ( m ) a λ 2 m +1 P a λ 2 m +1 P a λ 2 m +1 P a λ 2 m +1 P a M ( m ) - λ 2 m +1 M - λ 2 m +1 M T ( m ) a - - λ 2 m +1 T a λ 2 m +1 T a T able 9: Expanded generators in terms of the extended kinematical ones. By com bining the original comm utation relations of the extended kinematical algebras with the multiplication law of the semigroup S ( N ) E , one finds that the B k generalizations satisfy the comm utation relations listed in T able 10 . This generalization sho ws that b oth the extended kinematical algebras [ 51 , 92 ] and their Maxwellian extensions can be understo o d as particular sub cases of the B k extended kinematical algebra. F or N = 1, the resonant 0 S -reduced S ( N ) E - expansion reduces to a con traction corresp onding to the v anishing cosmological constant limit. In this case, the expanded algebras coincide with the B 3 algebra and its non-Loren tzian versions, namely the Poincar ´ e algebra and its non-degenerate non-Loren tzian coun terparts studied in [ 51 , 92 ] and listed in App endix A . In particular, the expanded generators are iden tified as J (0) ≡ J , G (0) a ≡ G a , H (0) ≡ H , P (0) a ≡ P a , S (0) ≡ S , T (0) a ≡ T a , C (0) ≡ C , M (0) ≡ M , B (0) ≡ B . (4.3) 16 Comm utators B k B k eB B k eC B k eS h J ( m ) , G ( n ) a i ϵ ab G ( m + n ) b ϵ ab G ( m + n ) b ϵ ab G ( m + n ) b ϵ ab G ( m + n ) b h J ( m ) , P ( n ) a i ϵ ab P ( m + n ) b ϵ ab P ( m + n ) b ϵ ab P ( m + n ) b ϵ ab P ( m + n ) b h G ( m ) a , G ( n ) b i − ϵ ab J ( m + n ) − ϵ ab S ( m + n ) − ϵ ab C ( m + n ) − ϵ ab B ( m + n ) h H ( m ) , G ( n ) a i ϵ ab P ( m + n ) b ϵ ab P ( m + n ) b ϵ ab T ( m + n ) b ϵ ab T ( m + n ) b h G ( m ) a , P ( n ) b i − ϵ ab H ( m + n ) − ϵ ab M ( m + n ) − ϵ ab H ( m + n ) − ϵ ab M ( m + n ) h H ( m ) , P ( n ) a i ϵ ab G ( m + n +1) b ϵ ab G ( m + n +1) b ϵ ab G ( m + n +1) b ϵ ab G ( m + n +1) b h P ( m ) a , P ( n ) b i − ϵ ab J ( m + n +1) − ϵ ab S ( m + n +1) − ϵ ab J ( m + n +1) − ϵ ab S ( m + n +1) h J ( m ) , T ( n ) a i - - ϵ ab T ( m + n ) b ϵ ab T ( m + n ) b h P ( m ) a , T ( n ) b i - - − ϵ ab C ( m + n +1) − ϵ ab B ( m + n +1) h C ( m ) , P ( n ) a i - - ϵ ab T ( m + n ) b ϵ ab T ( m + n ) b T able 10: Commutation relations of the B k generalization of the extended kinematical algebras. On the other hand, the case N = 2 repro duces the B 4 generalization of the kinematical algebra, whic h corresp onds precisely to the Maxwellian extended kinematical algebras discussed in the previous section. New families of kinematical symmetry algebras arise for N ≥ 3. These algebras are neither p ost-Newtonian nor p ost-Carrollian, and they cannot b e obtained as extensions of the Galilei or Carroll algebras. The particular case N = 3 repro duces the B 5 extended kinematical algebras, whose comm utation relations are listed in App endix B . At the relativistic level, the B 5 algebra has pro ven useful to recov er standard General Relativity from Chern-Simons and Born-Infeld gra vity actions [ 108 – 111 ]. Its non-relativistic non-degenerate coun terpart is given b y the B 5 eB algebra (see T able 14 ), introduced in [ 112 ] as a generalized Maxwellian exotic Bargmann algebra. In that work, its deriv ation was obtained through a con traction of the algebra B 5 ⊕ u (1) 4 . The Carrollian and static versions, which had not b een rep orted so far, are represented by the algebras B 5 eC and B 5 eS, resp ectiv ely , as display ed in T able 14 . F rom the commutation relations listed in App endix B , it is clear that the B 5 extended kinematical algebras arise as extensions of the Maxwellian extended kinematical algebras, whic h corresp ond to the first t wo rows of T able 14 . In particular, the first ro w repro duces the extended kinematical algebras in tro duced in [ 51 ]. The B k extended kinematical algebras admit the non-v anishing comp onents of the inv ariant tensor listed in T able 11 . The corresp onding coupling constan ts are related to those of the starting extended kinematical algebras through the semigroup elemen ts according to α i + j = λ i λ j µ r , β i + j = λ i λ j ν r , γ i + j = λ i λ j σ s , ζ i + j = λ i λ j ρ s (4.4) 17 with r = 0 , 1 and s = 0 , 1 , 2. Here, the product λ i λ j defined through the multiplication la w ( 4.1 ) of the semigroup S ( N ) E . In v ariant tensor B k B k eB B k eC B k eS ⟨ J ( m ) , J ( n ) ⟩ − α 2 n +2 m 0 − γ 2 n +2 m 0 ⟨ J ( m ) , H ( n ) ⟩ − α 2 n +2 m +1 0 − γ 2 n +2 m +1 0 ⟨ H ( m ) , H ( n ) ⟩ − α 2 n +2 m +2 0 − γ 2 n +2 m +2 0 ⟨ G ( m ) a , G ( n ) b ⟩ α 2 n +2 m δ ab β 2 n +2 m δ ab γ 2 n +2 m δ ab ζ 2 n +2 m δ ab ⟨ G ( m ) a , P ( n ) b ⟩ α 2 n +2 m +1 δ ab β 2 n +2 m +1 δ ab γ 2 n +2 m +1 δ ab ζ 2 n +2 m +1 δ ab ⟨ P ( m ) a , P ( n ) b ⟩ α 2 n +2 m +2 δ ab β 2 n +2 m +2 δ ab γ 2 n +2 m +2 δ ab ζ 2 n +2 m +2 δ ab ⟨ J ( m ) , S ( n ) ⟩ - − β 2 n +2 m - − ζ 2 n +2 m ⟨ J ( m ) , M ( n ) ⟩ - − β 2 n +2 m +1 - − ζ 2 n +2 m +1 ⟨ H ( m ) , M ( n ) ⟩ - − β 2 n +2 m +2 - − ζ 2 n +2 m +2 ⟨ H ( m ) , S ( n ) ⟩ - − β 2 n +2 m +1 - − ζ 2 n +2 m +1 ⟨ J ( m ) , C ( n ) ⟩ - - − γ 2 n +2 m 0 ⟨ J ( m ) , B ( n ) ⟩ - - - − ζ 2 n +2 m ⟨ P ( m ) a , T ( n ) b ⟩ - - γ 2 n +2 m +2 δ ab ζ 2 n +2 m +2 δ ab T able 11: Non-v anishing comp onents of the in v ariant tensor for the B k extended kinematical algebras. One can observ e that the comp onents prop ortional to α i , with i = 0 , . . . , N − 1, repro duce the in v ariant tensor structure of the B N +1 algebra. Since N = k − 2, this recursiv e prop ert y allows us to express the CS gravit y action for the B k algebra as the CS action for B k − 1 plus an additional con tribution prop ortional to α N : I B k = k 4 π Z  L B k − 1 + α N L N  . (4.5) An analogous recursiv e b ehavior holds for the non-Lorentzian counterparts of the B k algebra. 5 Discussion In this w ork, w e hav e presented a Maxwellian extension of the kinematical Lie algebras by promoting the contraction process of the original Bacry and L´ evy-Leblond cube [ 1 ] to an S-expansion mec hanism, with S (2) E as the relev ant semigroup. W e hav e sho wn that the non-degenerate non- Loren tzian Maxw ell gravit y theories previously constructed in the literature arise naturally within this unified framework. Moreov er, we ha ve shown that b oth the original cub e [ 1 ] and the Maxwellian coun terpart b elong to an infinite hierarch y of generalized kinematical algebras obtained through 18 arbitrary S ( N ) E semigroups, where the case N = 1 reproduces the Bacry and L ´ evy-Leblond cub e. These generalized structures corresp ond to the so-called B k algebras and their non-Loren tzian regimes. Remark ably , the present approac h also provides the non-v anishing comp onents of the in v ariant tensor required for the construction of CS gravit y actions for b oth Lorentzian and non- Loren tzian symmetry algebras. The results presen ted here op en several av en ues for further inv estigation. At the gravitational lev el, it w ould b e in teresting to study the dynamics associated with the Maxw ell kinematical algebras. In particular, it remains to clarify the ph ysical role of the additional gauge fields app earing in the non-Loren tzian regimes of the Maxwell cub e. These fields may admit an in terpretation in terms of p ost-Newtonian or p ost-Carrollian corrections, in analogy with higher-order expansions of relativistic gra vit y . It is also natural to explore whether the additional non-Lorentzian gauge fields can be related to gra vito-magnetic-like effects [ 113 ]. In this con text, one may ask whether they con tribute to generalized notions of torsion, non-inertial forces, or effective bac kground fluxes in non- relativistic and Carrollian geometries. F rom this p ersp ectiv e, the Maxwellian non-Lorentzian algebras in tro duced here ma y provide a no vel arena to in vestigate generalized gra vitational interactions b ey ond the standard kinematical frameworks. In the generalized kinematical setting, it w ould b e worth analyzing the space of classical solutions of the corresp onding B k CS gravities, including blac k-hole configuration, cosmological bac kgrounds, and their thermo dynamic prop erties. The additional conten t is exp ected to mo dify b oth the global structure of solutions and the asso ciated conserved charges, p oten tially leading to new thermo dynamic con tributions. The first non-trivial case beyond the Maxwellian lev el, corresp onding to the B 5 algebra, deserv es sp ecial attention. A t the relativistic level, B 5 has b een sho wn to play a central role in reco vering General Relativity in suitable limits [ 108 – 111 ]. It would b e highly interesting to in v estigate whether its non-Loren tzian coun terparts encode analogous subleading gra vitational structures and to elucidate the geometric and dynamical interpretation of the corresp onding gauge fields. On a more conceptual lev el, it w ould b e interesting to explore whether the non-Lorentzian algebras presen ted here admit realization as asymptotic symmetries of gravitational mo dels, and whether they giv e rise to new extensions of bms 3 algebra. On the other hand, motiv ated by recent dev elopments in three-dimensional AdS-Carroll gra vity , where the asymptotic symmetry algebra corresp onds to an infinite-dimensional extension of a generalized Maxw ell algebra [ 114 ], one could explore whether infinite-dimensional extensions of the B k algebras emerge in a symilar w ay . This could pro vide new insights into the role of non-Lorentzian limits within holography . Finally , another natural direction concerns the extension of our results to sup ersymmetric and higher-spin theories. In particular, applying the S -expansion metho d to non-Lorentzian regimes could lead to new classes of non-degenerate sup ergravit y theories. While sup ersymmetric extensions of the Bacry and L´ evy-Leblond cub e ha ve b een recently constructed in [ 92 ], it would b e in teresting to generalize the presen t approach b y considering different starting (sup er)algebras b eyond the (sup er) AdS case. A t the higher-spin lev el, it w ould be w orth exploring whether spin-5 / 2 symmetry algebras [ 115 – 118 ] can b e incorp orated in to generalized kinematical frameworks along the lines of [ 119 ]. 19 Ac kno wledgmen ts This w ork was funded by the National Agency for Research and Developmen t ANID - F ONDE- CYT gran ts No. 1250642 and 1231133. This w ork was supp orted b y DIREG 04/2025 of the Univ ersidad Cat´ olica de la San t ´ ısima Concep ci´ on (P .C.). P .C. and E.R. would like to thank the Direcci´ on de Inv estigaci´ on and Vice-rector ´ ıa de Inv estigaci´ on of the Univ ersidad Cat´ olica de la San t ´ ısima Concep ci´ on, Chile, for their constant supp ort. A Explicit comm utation relations of extended kinematical algebras This appendix con tains the tables with the complete list of comm utation relations of the extended kinematical algebras app earing in [ 51 ]. Comm utators AdS P oincar´ e Extended Extended Newton-Ho ok e Bargmann [ J, G a ] ϵ ab G b ϵ ab G b ϵ ab G b ϵ ab G b [ J, P a ] ϵ ab P b ϵ ab P b ϵ ab P b ϵ ab P b [ G a , G b ] − ϵ ab J − ϵ ab J − ϵ ab S − ϵ ab S [ H , G a ] ϵ ab P b ϵ ab P b ϵ ab P b ϵ ab P b [ G a , P b ] − ϵ ab H − ϵ ab H − ϵ ab M − ϵ ab M [ H , P a ] ϵ ab G b 0 ϵ ab G b 0 [ P a , P b ] − ϵ ab J 0 − ϵ ab S 0 T able 12: Commutation relations of the AdS, P oincar´ e, extended Newton-Ho oke and extended Bargmann algebras. Comm utators Extended Extended Extended Extended AdS-Carroll Carroll AdS-Static Static [ J, G a ] ϵ ab G b ϵ ab G b ϵ ab G b ϵ ab G b [ J, P a ] ϵ ab P b ϵ ab P b ϵ ab P b ϵ ab P b [ G a , G b ] − ϵ ab C − ϵ ab C − ϵ ab B − ϵ ab B [ H , G a ] ϵ ab T b ϵ ab T b ϵ ab T b ϵ ab T b [ G a , P b ] − ϵ ab H − ϵ ab H − ϵ ab M − ϵ ab M [ H , P a ] ϵ ab G b 0 ϵ ab G b 0 [ P a , P b ] − ϵ ab J 0 − ϵ ab S 0 [ J, T a ] ϵ ab T b ϵ ab T b ϵ ab T b ϵ ab T b [ P a , T b ] − ϵ ab C 0 − ϵ ab B 0 [ C, P a ] ϵ ab T b ϵ ab T b ϵ ab T b ϵ ab T b T able 13: Commutation relations for the extended versions of the AdS–Carroll, Carroll, AdS–Static, and Static algebras. 20 B Explicit comm utation relations of the B 5 extended kinematical algebras This app endix con tains the table with the complete list of commutators of the B 5 algebra and its non-Loren tzian counterpart. Here, the generators hav e b een identified in terms of the expanded Comm utators B 5 B 5 eB B 5 eC B 5 eS [ J , G a ] ϵ ab G b ϵ ab G b ϵ ab G b ϵ ab G b [ J , P a ] ϵ ab P b ϵ ab P b ϵ ab P b ϵ ab P b [ G a , G b ] − ϵ ab J − ϵ ab S − ϵ ab C − ϵ ab B [ H , G a ] ϵ ab P b ϵ ab P b ϵ ab T b ϵ ab T b [ G a , P b ] − ϵ ab H − ϵ ab M − ϵ ab H − ϵ ab M [ J , T a ] - - ϵ ab T b ϵ ab T b [ C , P a ] - - ϵ ab T b ϵ ab T b [ H , P a ] ϵ ab Z b ϵ ab Z b ϵ ab Z b ϵ ab Z b [ P a , P b ] − ϵ ab Z − ϵ ab T − ϵ ab Z − ϵ ab T [ J , Z a ] ϵ ab Z b ϵ ab Z b ϵ ab Z b ϵ ab Z b [ Z , G a ] ϵ ab Z b ϵ ab Z b ϵ ab Z b ϵ ab Z b [ G a , Z b ] − ϵ ab Z − ϵ ab T − ϵ ab L − ϵ ab Y [ P a , T b ] - - − ϵ ab L − ϵ ab Y [ J , N a ] ϵ ab N b ϵ ab N b ϵ ab N b ϵ ab N b [ Z , P a ] ϵ ab N b ϵ ab N b ϵ ab N b ϵ ab N b [ H , Z a ] ϵ ab N b ϵ ab N b ϵ ab L b ϵ ab L b [ Z a , P b ] − ϵ ab N − ϵ ab V − ϵ ab N − ϵ ab V [ G a , N b ] − ϵ ab N − ϵ ab V − ϵ ab N − ϵ ab V [ J , L a ] - - ϵ ab L b ϵ ab L b [ Z , T a ] - - ϵ ab L b ϵ ab L b [ C , N a ] - - ϵ ab L b ϵ ab L b [ L , P a ] - - ϵ ab L b ϵ ab L b T able 14: Commutation relations of the B 5 extended kinematical algebras. generators listed in T able 9 as J (0) ≡ J , J (1) ≡ Z , G (0) a ≡ G a , G (1) a ≡ Z a , H (0) ≡ H , H (1) ≡ N , P (0) a ≡ P a , P (1) a ≡ N a , S (0) ≡ S , S (1) ≡ T , T (0) a ≡ T a , T (1) a ≡ L a , C (0) ≡ C , C (1) ≡ L , B (0) ≡ B , B (1) ≡ Y , M (0) ≡ M , M (1) ≡ V . 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