Newton-Hooke/Carrollian expansions of (A)dS and Chern-Simons gravity

We construct finite- and infinite-dimensional non-relativistic extensions of the Newton-Hooke and Carroll (A)dS algebras using the algebra expansion method, starting from the (anti-)de Sitter relativistic algebra in D dimensions. These algebras are also shown to be embedded in different affine Kac-Moody algebras. In the three-dimensional case, we construct Chern-Simons actions invariant under these symmetries. This leads to a sequence of non-relativistic gravity theories, where the simplest examples correspond to extended Newton-Hooke and extended (post-)Newtonian gravity together with their Carrollian counterparts.


Introduction
Non-relativistic symmetries are usually obtained from relativistic ones by means of Lie algebra contractions. The most well-known example is the Wigner-Inönü contraction of the Poincaré algebra that leads to the Galilei algebra [1] by sending the speed of light to infinity. Lie algebra contractions necessarily preserve the number of generators but can alter the cohomology of the Lie algebra, thus allowing for different central extensions like the Bargmann extension of the Galilei algebra [2], which is crucial when taking the limit of the relativistic particle action to the non-relativistic one [3,4].
In recent years, more general constructions of non-relativistic symmetry algebras from relativistic ones have been explored. One example is given by the method of Lie algebra expansions [5][6][7] applied to the Poincaré algebra [8]. This method provides an infinite sequence of non-relativistic algebras extending the Galilei algebra with an increasing number of generators, which have been used in [8][9][10][11] to construct various gravitational actions. The Lie algebra expansion method can also be related to a sequence of post-Newtonian limits as shown in [12], and has also been applied to derive diverse non-relativistic symmetries in the context of (super-)gravity [13][14][15][16][17]. Another method is based on a Galilean free Lie algebra [18] that can be thought as the most general extension of the Galilei algebra JHEP02(2020)009

Expansions of the (A)dS algebra
In this section, we will construct an infinite family of non-relativistic algebras of the Newton-Hooke and Carroll AdS type. The first members of this family will be given by the Newton-Hooke algebra in the former case and the Carroll AdS algebra in the latter (without extensions) [19,20]. We also obtain an infinite-dimensional algebra that contains these and other intermediate cases, generalises the one obtained in [8,10,18], as quotients. All our algebras contain a cosmological constant as we start from the AdS algebra.
We will construct the series of non-relativistic algebras by means of Lie algebra expansions [5][6][7]. More precisely, we will use the semigroup expansion technique [7] with semigroup S   We note that the decomposition (2.2) is adapted to point particles in the sense that one direction -that can be thought of as the world-line direction of the particle -is singled out. In appendix A.2, we also consider the case of extended objects.

Extended Newton-Hooke algebras
In order to perform expansions of (A)dS of Newton-Hooke type, we first note that (2.3) allows for the following subspace decomposition V = V 0 ⊕ V 1 [8] V 0 = {H,J ab } ,

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which satisfies a Z 2 -graded structure, i.e., This decomposition is similar to a symmetric space decomposition. The homogeneous coset space in the case of AdS is SO(D − 1, 2)/(SO(D − 1) × R) and SO(D, 1)/(SO(D − 1) × R) in the dS case. We will consider expansions with the semigroup S (N ) E = {λ 0 , . . . , λ N +1 }, whose multiplication law is given by [7] (2.6) where λ N +1 acts as the zero of the semigroup, as it satisfies for all λ i . This semigroup admits the subset decomposition which is compatible with (2.5) in the sense that and therefore resonant with the choice of V 0 and V 1 in (2.4). Thus, a reduced resonant expanded algebra can be defined as the direct sum where the reduction condition in the algebra is implemented by the constraints which map the zero λ N +1 of the semigroup to the zero element in the expanded algebra. In the following we will show how, for different choices of the semigroup S (N ) E , the reduction (2.9) lead to a non-relativistic algebras that generalise the Newton-Hooke symmetry.
The simplest example in our construction corresponds to the expansion with the semigroup S (1) E , which as we will see is equivalent to a non-relativistic contraction of the (A)dS algebra. In other words, this expansion gives the Newton-Hooke algebras. By setting N = 1 in (2.6) and (2.9), the generators of the expanded algebra are given by (2.11)

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Using the relativistic commutation relations (2.3) together with the reduction condition (2.10) for the zero element λ 2 , the commutation relations for the generators (2.11) read which corresponds to the Newton-Hooke ∓ algebra (with Λ ≶ 0) without extensions [19,20], this algebra reduces to the Galilei symmetry for Λ = 0. Therefore the use of the semigroup S (N ) E will create a family of generalised Newton-Hooke algebras for all the possible values of N .
The case with N = 2 leads to a Newton-Hooke algebra with some extensions. Unlike the previous case, when using the semigroup S (2) E , the element λ 2 is no longer the zero of the semigroup. In this case the reduction condition (2.10) holds for a new semigroup zero element λ 3 , and (2.11) has to be supplemented with two extra expanded generators Using the semigroup product law (2.6) for N = 2, we find that the non-vanishing commutation relations for the expanded algebra are given by the Newton-Hooke commutators (2.12) together with This expansion produces the Bargmann central extension M and a non-central extension S ab [40]. This algebra can be understood as the generalisation of the double central extension [41][42][43] of the Newton-Hooke algebra to D > 3. In fact, in 2+1 dimensions the generator S ab can be dualised to a scalar and becomes central. In that case the algebra (2.14) becomes exactly the extended Newton-Hooke algebra (Λ = 0) [44,45] or the extended Bargmann algebra (Λ = 0) [46,47].
Extended (post-)Newtonian gravity algebra. For N = 3 we obtain the Newton-Hooke version of the algebra found in [9] as the symmetry of post-Newtonian gravity. Moreover, when considering N = 4, we find an extension of the post-Newtonian gravity algebra that generalises the algebra of [11] to any dimension. Explicitly, for N = 3 the generators of the expanded algebra are given by (2.11), (2.13) and (2.15) The zero of the semigroup S E is λ 4 , and the reduction condition (2.10) together with the semigroup law and the (A)dS commutation relations (2.3) leads to the following non-JHEP02(2020)009 vanishing commutators for the expanded algebra: In the case Λ = 0 this algebra corresponds to the one found in [9] and further studied in [8,12,18]. One can extend this algebra by considering N = 4 in the expansion prescription (2.9). In this case, (2.15) has to be supplemented with extra expanded generators given by The zero element in S E is given by λ 5 , which means that the reduction condition (2.10) in this case sets λ 5 ⊗J AB = λ 5 ⊗P A ≡ 0 and the non-vanishing commutation relations of the corresponding expanded algebra are given by (2.16) In the case D = 2 + 1 and Λ = 0 this algebra corresponds to the one studied in [11] and defines a central extension of the post-Newtonian gravity symmetry (2.18). For D > 2 + 1, however, the generator Z ab is no longer central, exactly as it happens with S ab in the extension of the Newton-Hooke algebra. Notice that if some generators are central for some value N of the semigroup, they are no longer central in the N + 1 case. More specifically, the expansions with odd N do not possess central elements, while expansions with even values of N always yield two central elements given by λ N ⊗J and tilde λ N ⊗H.
Instead of continuing with increasing values of N , in the next subsection we will construct an infinite-dimensional graded algebra using an infinite-dimensional semigroup.
Infinite-dimensional Galilean algebra. The previous analysis suggests that we can construct an infinite-dimensional graded non-relativistic algebra using the expansion procedure with an infinite-dimensional semigroup of the form [14] S (∞) = {λ 0 , λ 1 , λ 2 , . . . } (2.19) JHEP02(2020)009 with multiplication law λ α · λ β = λ α+β . (2.20) (This semigroup can be realised as powers of a formal variable ω by letting λ α = ω α . The finite semigroups S (N ) E then correspond to working to order O(ω N +1 ).) As this semigroup does not have a zero element, the resonant subset decomposition that satisfies (2.8) in this case is simply given by S 21) and the corresponding resonant (non-reduced) expansion of (A)dS is where V 0 and V 1 were defined in (2.4). Denoting the generators of the expanded algebra (2.22) as J leads to the infinite-dimensional graded algebra , , We can get the finite-dimensional algebras presented in the previous subsections as quotients of (2.24) by suitable infinite ideals. For example, in the case of (2.16) the ideal is generated by J a , H (m) for m > 2. Taking the limit Λ → 0, we obtain as a contraction of (2.24), the infinite-dimensional extension of the Galilei algebra introduced in [10] and further studied in [18].

Extended Carroll (A)dS algebras
The (A)dS Carroll algebra corresponds to the ultra-relativistic contraction of the (A)dS algebra (2.1) [19]. It can alternatively be obtained by means of an expansion procedure when considering the following subspace decomposition for (A)dS, which is Z 2 -graded. Note that here P a and H have been interchanged with respect to the subspace decomposition used in the Newton-Hooke case (2.4). This is a special case of a general duality at the level of the translation generators between Galilean and Carrollian symmetries [24]. Using the decomposition (2.25), we can generalise the procedure outlined in the previous section to define Carrollian expansions of (A)dS, whose simplest JHEP02(2020)009 case is the Carroll (A)dS algebra. Indeed, the non-vanishing commutation relations of the corresponding reduced resonant expanded algebra is given by are given in (2.7), and the case N = 1 leads to the (A)dS Carroll algebra in D dimensions where the expanded generators have been defined as Naturally, when setting Λ = 0, this allows one to obtain the Carroll algebra in D dimensions as an expansion of the Poincaré algebra. Subsequently, we can consider greater values of N to obtain extended Carroll (A)dS algebras. In the case N = 2 the expanded algebra has extra generators given by and the commutation relations are the ones of (2.27) together with In the case N = 3, we get two extra generators, The commutation relations of this expanded algebra are (2.27), (2.30) plus For Λ = 0, this algebra defines a Carrollian counterpart of the (post-)Newtonian symmetry introduced in [10] in the context of general relativity. In the same way, one can check that the N = 4 case defines a higher-dimensional Carrollian (A)dS analogue of the extended (post-)Newtonian symmetry given in [11].

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Infinite-dimensional Carroll algebra. Similarly to the Newton-Hooke construction, the ultra-relativistic expansions of (A)dS form a family of algebras, which can be described in a unified fashion by considering the infinite semigroup (2.19) and the non-reduced resonant expansion S are given in (2.21). By defining an infinite set of expanded generators by J we obtain an infinite-dimensional Carrollian expansion of the (A)dS algebra , , The different finite expansions previously constructed using the semigroup S can be obtained from the infinite case by considering suitable quotients. Redefining the generators according to and, taking the limit Λ → 0, we obtain an extension of the infinite-dimensional algebra in [10] as a contraction of the infinite-dimensional extended Carrollian algebra (2.35). A related contraction was obtained from the infinite-dimensional extended Newton-Hooke algebra (2.24) above. In the contraction of the Carroll algebra (2.35), it is extended in a semidirect sum by the additional generator P (0) a (which should then rather be called P (−1) a in the notation of [10]). Conversely, the algebra in [10] can be seen as an extension of the contracted infinite-dimensional extended Carroll algebra if we again redefine H (m) → H (m+1) and then add a generator H (0) .

Newton-Hooke and Carrollian affine algebra
In this section, we will show that the infinite-dimensional Lie algebras (2.24) and (2.35) also can be obtained from the extension of so(D − 1) to an (untwisted or twisted) affine Kac-Moody algebra. We shall only require parabolic subalgebras of these Kac-Moody algebras and this construction can be linked to free Lie algebras in terms of quotients by Serre relations. The algebras for finite N discussed in the previous section then correspond to further quotients, similar to the constructions in [18,48]. JHEP02(2020)009

Newton-Hooke affine algebras
Consider first the complex Lie algebra . Thus g r is the complexification of so(d). We can extend g r to g r+1 (either D r+1 or B r+1 ) by adding a node labelled 0 and then further to an affine Kac-Moody algebra g (1) r+1 by adding a node labelled −1 to the Dynkin diagram of g r . The resulting diagram, together with our labelling of the nodes, is shown in figure 1. The corresponding Cartan matrix is for B (1) r+1 (where rows and columns are counted −1, 0, 1, . . . , r from left to right and from top to bottom).
When we add the two nodes 0 and −1 we also add six generators e i , f i , h i for i = 0, −1 to the generators of g r . This gives a central extension of the loop algebra of g r+1 where JHEP02(2020)009 the central element is given by The affine algebra g (1) r+1 is then obtained from it by adding also a derivation generator d satisfying [d, e −1 ] = e −1 and [d, f −1 ] = −f −1 and commuting with all the other generators.
We consider the subalgebra of g (1) r+1 generated by e i , h i for i = 0, −1 together with the generators of g r . This Lie algebra has an (N × N)-grading associated to these two nodes, which can be trivially extended to a (Z × Z)-grading. It can thus be decomposed into a direct sum of subspaces, each labelled by a pair ( 0 , −1 ) of non-negative integers. The subspace labelled by ( 0 , −1 ) is spanned by elements formed as multibrackets of the generators, where e 0 and e −1 appear 0 and −1 times, respectively. The subalgebra at ( 0 , −1 ) = (0, 0) contains g r but also the two additional Cartan generators h 0 and h −1 . By taking the linear combinations (such that c = h + h ) we get elements that commute with g r . Furthermore, they satisfy The (Z × Z)-grading gives rise to a Z-grading, where the single level is the sum of 0 and −1 . Since g r is a subalgebra at level zero, we get representations of it at each level. At level = 0 we have the generators J ab of (the complexification of) so(d) and the two so(d) scalars h and h . At level = 1 we have two lowest weight representations with lowest weight vectors e −1 and e 0 , respectively. The Dynkin labels of both corresponding highest weight representations are [1, 0, 0, . . . , 0] since Thus they are vector representations, and we denote the corresponding generators by X a (with lowest weight vector e −1 ) and Y a (with lowest weight vector e 0 ), where a = 1, 2, . . . , d as before. It follows from (3.5) that (The parabolic subalgebra of the 'horizontal' algebra g r+1 is generated by J ab , Y a and h 0 .) In the free Lie algebra generated by all X a and Y a at level = 1, the subspace at level = 2 decomposes into a direct sum of so(d) modules with Dynkin labels

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(where ∧ denotes the antisymmetric tensor product, coming from the antisymmetry of the Lie bracket). The free Lie algebra construction continues to all positive levels [18,48] but in order to reproduce the algebra (2.24) we need to take a quotient. This quotient leads to the subalgebra of g (1) r+1 at positive levels and the ideal that one quotients out is generated by the Serre relations at level = 2. We denote the corresponding 2-form and scalar generators by J ab 2 and h 2 , respectively, where the superscript indicates that they appear at level = 2. The commutation relations among the generators at level = 1 giving rise to these generators at level = 2 are The pattern with two vectors at odd levels and a 2-form and a scalar at even levels continues, as shown in for k ≥ 0 (where again the superscript is the level ), and the non-vanishing commutation relations (except for those involving h , which we have omitted since they turn out to be irrelevant) are JHEP02(2020)009 where J ab 0 = J ab , h 0 = h, X a 1 = X a and Y a 1 = Y a . If we then set for Λ > 0, then we recover the commutation relations (2.16). More generally, if we set for Λ > 0, then we recover the commutation relations (2.24). These formulas are still valid for Λ < 0 if we interpret √ Λ as ±i |Λ| and we will then obtain a different real form of the complex Lie algebra g (1) r+1 .

(A)dS Carrollian affine algebras
The procedure for obtaining (2.35) from an affine algebra is very similar to the previous discussion so we shall be rather brief.
We first note that due to the choice of subspaces in (2.25), the infinitely expanded algebra (2.35) consists of infinite repetitions of copies of {J ab ,P a } as even level spaces and {H,G a } as odd level spaces, see (2.33). Since in the Carrollian limit the commutation relations (2.27) are perfectly compatible with identifying the complexification of V 0 = {J ab ,P a } as the algebra so(D) and the complexification of V 1 = {H,G a } as its D-dimensional vector representation, we have to look for a Kac-Moody algebra where these two spaces repeat infinitely. Thus, there must be Serre relations corresponding to an ideal such that we obtain the positive levels as a quotient of the free Lie algebra generated by the space V 1 . The problem turns out to be identical to one already solved in [18], where the corresponding algebras were identified as the untwisted affine algebra of type B

Chern-Simons non-relativistic gravities
In this section we want to find realisations of the previous non-relativistic symmetries. We will be interested in the construction of Chern-Simons gravities in 2+1 dimensions, defined by the action 5 where the gauge algebra is given by (2.24) and (2.35). In order to carry out this construction we shall construct invariant tensors of these algebras in the following way: in 2 + 1 dimensions, an invariant tensor for the expanded algebras of interest can be obtained following [7], by first defining the structure constants K γ αβ of the semigroup (2.19) as Given the invariant tensor on so(2, 2),

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the invariant tensor on the expanded algebra can be defined as where we have introduced an infinite set of arbitrary constants α (γ) .

Extended Newton-Hooke gravities
Now we will consider the infinite-dimensional extended Newton-Hooke algebra (2.24), where the definition (2.23) of the expanded generators leads to Therefore, as the constants α (γ) in (4.5) are non-vanishing only for even values of γ, it is convenient to relabel them in terms of a set of constants µ (m) = α (2m) . Finally, using (2.23) and dualising J ab and B a in the form In terms of the dual generators (4.6), the infinite-dimensional graded algebra (2.24) takes the form  where the curvature components can be worked out using the algebra (4.8) and read explicitly (4.11) Here and from now on the wedge product between forms are not written explicitly. Using the gauge connection (4.9) and the invariant bilinear form (4.7), the action (4.1) takes the form where the different terms can be rearranged in terms of µ (n) as We shall now discuss in more detail the first three terms in the sum.
• The µ (0) term can be directly read off from the second sum in (4.12). Up to total derivatives it reads which corresponds to Galilean gravity in 2+1 dimensions [49].
• In order to write down the µ (1) term in a familiar way, we first note that we can relate the curvatures associated to the different generators (4.11) in the form , where T.D. stands for total derivative terms. Using these relations, the action S (1) takes the form This action defines extended Newton-Hooke gravity [25,26] and generalises extended Bargmann gravity [27,28] to include the cosmological constant. This can be seen clearly by relabeling the gauge fields as • For the µ (2) term we use the relations , 18) which allows one to express S (2) as

(4.19)
For Λ = 0, this is precisely the Lagrangian for non-relativistic three-dimensional gravity found in [11], where the gauge fields should identified as in (4.20) together with (4.20) In general the action S (i) can be written, after partial integration, in the form

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It is important to note that the infinite sum (4.13) can be consistently truncated to give which gives an action for the gauge fields e a (m) , ω a (m) , for m = 0, 1, . . . , i 0 , τ (n) , ω (n) , for n = 0, 1, . . . , i 0 + 1 , (4. 23) and sets all the other gauge fields to zero. In this case, the invariant tensor (4.7) in nondegenerate for µ (i 0 +1) = 0, and the truncation leads to the condition This corresponds to the Chern-Simons action for non-relativistic gravity invariant under the expanded algebra (2.9) for N = 2(i 0 + 1), this algebra can be written as the direct sum L The invariant tensor in this basis follows from (4.7) and has takes the form This isomorphism implies that one can reformulate the expanded Newton-Hooke gravity action (4.12) in the form

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where the gauge connections are given by This is in completely analogy with the relativistic case, where three-dimensional Einstein gravity with negative cosmological constant can be reformulated as a Chern-Simons theory of the form (4.30) with chiral connections taking values in the sl(2, R) algebra [50][51][52][53][54]. In fact, the algebra (4.32) can be obtained as a non-relativistic S (∞) expansion of so(2, 1) ≈ sl(2, R) by defining The case N = 1 shows that the Newton-Hooke algebra in 2+1 dimensions is isomorphic to two copies of the Euclidean algebra in 1+1 dimensions, while the N = 2 case expresses the extended Newton-Hooke symmetry in three-dimensions as two copies of the centrally extended Newton-Hooke symmetry in two dimensions [45], isomorphic to the Nappi-Witten algebra [16,55].
Exotic invariant bilinear form. As so(2, 2) is semi-simple there are two independent invariant bilinear forms. Besides (4.3), a second invariant bilinear form on so(2, 2) is which induces the following invariant tensor for the expanded generators (2.23) Defining β (2m) = ν (m) and using (4.6) this leads to

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In this case, the action (4.1) boils down to The exotic invariant tensor in the chiral basis (4.27) takes the form Therefore, the action (4.40) can be reformulated in terms of (4.31) as where now the relative sign between the two Chern-Simons actions is changed compared to (4.30).

Extended Carrollian gravities
In the Carrollian case, the definition of the expanded generators with algebra given in (2.35) together with (4.4) yields the following invariant tensor Thus, in contrast with the Newton-Hooke case, the constants α (γ) in (4.5) are non-vanishing only for odd values of γ, and we will relabel them as ρ (m) = α (2m+1) . Using (2.34) and the dual generators (4.6) the invariant bilinear form in the D = 3 case can also be written as The infinite-dimensional algebra (2.35) takes the form

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Action. In the Carrollian case, the connection one-form (4.9) together with the commutation relations (4.45) yields a curvature two-form (4.10) with the following components: (4.46) The Chern-Simons action (4.1) then takes the form (4.47) which can be also written as where after integration by parts we can write S (i) as (4.49) The first action in this sequence is given by where we have used the field redefinition (4.17). This corresponds to Carrollian AdS Chern-Simons gravity [39,56] and reduces to three-dimensional Carroll gravity [43,57,58] in the vanishing cosmological constant limit. 6 Using also (4.20), the second term of the sum reads The first realisations of the Carroll symmetry in gravity appeared in the study of the zero signature limit [59,60] or strong coupling limit [61,62] (see also [63,64]) of general relativity, the last one being closely related to the Belinski-Khalatnikov-Lifshitz (BKL) limit of gravity [65][66][67]. On the other hand, the ultra-relativistic expansion of general relativity has been explored in [68], while the Carrollian limit at the level of the Einstein-Hilbert action has been studied in [49]. It is also important to mention the realisation of the Carroll symmetry in the near horizon limit of black holes [69].

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and defines a Carrollian counterpart of (post-)Newtonian gravity action [9] in three dimensions including cosmological constant. Similarly, the next actions in the sequence define further post-Carrollian corrections for three-dimensional Carroll (A)dS gravity. In is interesting that unlike the Newton-Hooke expansion, truncations of the general Carrollian action (4.48) for arbitrary r lead to a non-degenerate invariant bilinear form for the corresponding truncation in (4.45). Also, the extended Carrollian algebras for (A)dS do not include central terms, which is in contrast with the Newton-Hooke case, where the central extensions are precisely the ones that allow us to find non-generate pairings when (4.22) holds.
Exotic invariant bilinear form. We can also consider the exotic invariant tensor for (A)dS (4.35), which leads to the following expression for pairings of the expanded generators (2.34) Defining β (2m) = σ (m) and using (4.6) this can also be written as In this case, the action (4.1) reduces to (4.55) One can consider again finite truncations of this actions and, which produce exotic Carroll and post-Carrollian gravity theories in 2 + 1 dimensions.

Conclusions and outlook
We have studied the non-relativistic symmetries obtained by Lie algebra expansion of the AdS or dS algebra in D space-time dimensions, i.e. so(D − 1, 2) or so(D, 1). This generalises previous constructions to include a cosmological constant and generates an infinite family of algebras of Newton-Hooke [19,20]  expansions contained in each family can be systematically constructed as a free algebra and correspond to suitable quotients of the infinite-dimensional case.
We have considered a gravitational model based on them by focussing on the case of (2 + 1)-dimensional Chern-Simons theories. We show that the family of algebras generates systematically non-relativistic gravity theories extended by a cosmological constant, such as extended Bargmann gravity [26,28] and (post-)Newtonian gravity [11].
As stated in [10] these extended Newton-Hooke symmetries encode the large c expansion of general relativity. An interesting way to see this in 2+1 dimensions is that the Chern-Simons formulation of AdS three-dimensional gravity allows to construct the metric out of the chiral connections (4.31) in the form [70] Using (4.31) and rearranging the sum in terms of γ (n) we get The flat version of this metric has been studied in [12] as an infinite extension of Minkowski space where post-Newtonian corrections to relativistic symmetries can be naturally implemented. This metric can be found in the vanishing cosmological constant limit by defining coordinates x µ (m) such that τ (m) = τ (m)µ dx µ (m) and e a (m) = e a (m)µ dx µ (m) . Then, the Minkowskian case is obtained by setting e a (m)µ = δ a µ and τ (m)µ = (1, 0, 0, 0) for all m, which leads to the expanded metric constructed in [12].
On the other hand, keeping the gauge field components τ (m)µ and e a (m)µ arbitrary while identifying the coordinates as x µ (m) = x µ , allows one to find the (2+1)-dimensional version of the post-Newtonian expansion of the metric used in [10,[29][30][31][32][33]. Indeed, identifying the invariant tensor constants as powers of the speed of light in the form γ (n) → c 2(1−n) , and defining the spatial metrics h µν dx µ dx ν = e a (0) e a(0) and Φ µν dx µ dx ν = e a (0) e a(1) + e a (1) e a(0) , this expression reproduces the aforementioned non-relativistic expansion: where we have also relabeled the metric fields according to (4.17) and (4.20). Alternatively one can introduce an expansion of the dreibein forms

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which also leads to the expansion of the metric (5.2) with γ (n) = ν (n) , where ν (n) appears in the pairing of the momentum generators in (4.39).
We have also considered the Carrollian case with cosmological constant to construct a corresponding Chern-Simons theory based on the algebra (2.35). This reproduces known ultra-relativistic gravity theories in 2+1 dimensions and an infinite family of generalisations.
Applying the same procedure as (5.2) in the case of expanded Carrollian (A)dS algebras (4.45) leads to a novel expansion for the metric tensor that reads where we have also defined Ψ µν dx µ dx ν = e a (1) e a(1) + e a (0) e a(2) + e a (2) e a(0) . This metric can be conjectured to describe the ultra-relativistic expansion of three-dimensional Einstein gravity. To evaluate whether this expansion can be generalised to higher dimensions in the context of the Carroll limit of general relativity along the lines of [59] or [68] is an interesting question that we hope to address in the future.
There are several directions in which this research could be extended. One would be to generalise our work of symmetries of post-Newtonian correction in the flat Minkowski space [12] to curved (A)dS space and construct particle actions in the curved generalisation of the infinite-dimensional Minkowski space. On the other hand it would be interesting to analyse particle systems in the presence of a constant background field by considering the non-relativistic expansions of the Maxwell algebra according to appendix A.1. One could also construct actions for extended objects by means of the p-brane symmetries outlined in appendix A.2. Finally, it would be interesting to extend the analysis to the supersymmetric (A)dS case.

A Further generalisations
The procedure for obtaining non-relativistic expansions of a given relativistic algebra can be generalised to several other interesting cases. Here, we apply the general scheme to obtain non-relativistic Maxwell algebras and non-relativistic brane symmetries.

A.1 Non-relativistic expansions of the Maxwell algebra
The relativistic Maxwell algebra in D space-time dimensions is given is given by the following extension of the Poincaré algebra [72] (see also [48,73,74]  algebra [18], while the N = 2 case leads to a generalisation of the exotic Maxwellian Bargmann algebra [75] for D > 3. For N = 3, the resulting algebra is a Maxwell extension of the post-Newtonian symmetry (2.24) with Λ = 0, first found in [10]. For greater values of N we obtain further post-Newtonian corrections of the electric non-relativistic Maxwell algebra.
Carrollian expansions. Similarly, we can define Carrollian expansions of the Maxwell algebra by using the following alternative Z 2 -graded subspace decomposition where we have interchanged the generators H and P a with respect to the Galilean case. This leads to the resonant non-reduced expanded algebra