Three-dimensional Spin-3 Theories Based on General Kinematical Algebras

We initiate the study of non- and ultra-relativistic higher spin theories. For sake of simplicity we focus on the spin-3 case in three dimensions. We classify all kinematical algebras that can be obtained by all possible In\"on\"u--Wigner contraction procedures of the kinematical algebra of spin-3 theory in three dimensional (anti-) de Sitter space-time. We demonstrate how to construct associated actions of Chern--Simons type, directly in the ultra-relativistic case and by suitable algebraic extensions in the non-relativistic case. We show how to give these kinematical algebras an infinite-dimensional lift by imposing suitable boundary conditions in a theory we call"Carroll Gravity", whose asymptotic symmetry algebra turns out to be an infinite-dimensional extension of the Carroll algebra.


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1 Introduction Due to the principle of relativity, the notion of kinematical or space-time symmetry algebras, which contain all symmetries that relate different inertial frames, is a crucial ingredient in the construction of physical theories. Bacry and Lévy-Leblond have classified all possibilities for kinematical algebras [1], consisting of space-time translations, spatial rotations and boosts, under some reasonable assumptions. Apart from the relativistic Poincaré and (A)dS algebras, this classification also contains the Galilei and Carroll algebras (and generalizations thereof that include a cosmological constant), that appear as kinematical algebras in the non-relativistic (c → ∞) and ultra-relativistic (c → 0) limit. Even though fundamental theories are relativistic, the Galilei and Carroll algebras continue to play an important role in current explorations of string theory and holography. For instance, non-relativistic symmetries underlie Newton-Cartan geometry, a differential geometric framework for non-relativistic space-times that has found recent applications in holography [2][3][4][5][6][7][8][9][10], Hořava-Lifshitz gravity [11][12][13] and in the construction of effective field theories for strongly interacting condensed matter systems [14][15][16][17][18][19][20][21].

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On the other hand, ultra-relativistic Carroll symmetries have recently been studied in relation to their connection [22] with the Bondi-Metzner-Sachs (BMS) algebra of asymptotic symmetries of flat space-time [23,24]. As such, Carroll symmetries play a role in attempts to construct holographic dualities in asymptotically flat space-times [25][26][27][28][29][30][31][32][33], as symmetries of the S-matrix in gravitational scattering [34] and in the recent notion of 'soft hair' on black hole horizons [35,36]. 1 The kinematical algebras that have been classified by Bacry and Lévy-Leblond pertain to theories that contain bosonic fields with spins up to 2. One can also consider theories in which massless higher spin fields are coupled to gravity [42]. These so-called 'higher spin gauge theories' have been formulated in (A)dS space-times (see [43][44][45] for reviews) and have featured prominently in the AdS/CFT literature, as a class of theories for which holographic dualities can be constructed rigorously [46][47][48][49][50][51][52][53][54], essentially because they are a 'weak-weak' type of duality, i.e., CFTs with unbroken higher spin currents are free [55]. They typically contain an infinite number of higher spin fields. As a consequence, their space-time symmetries are extended to infinite-dimensional algebras that include higher spin generalizations of space-time translations, spatial rotations and boosts. Higher spin gauge theories have thus far mostly been considered in relativistic (A)dS space-times, with relativistic CFT duals. 2 Since both higher spin gauge theories as well as non-and ultra-relativistic space-time symmetries have played an important role in recent developments in holography, it is natural to ask whether one can combine the two. In order to answer this question, one needs to know which non-and ultra-relativistic kinematical algebras can appear as spacetime symmetries of higher spin theories. This is the problem that we will start addressing in this paper, in the context of higher spin gauge theories in three space-time dimensions.
The reason for restricting ourselves to three space-time dimensions stems from the fact that, as far as higher spin gauge theory is concerned, this case is a lot simpler than its higherdimensional counterpart. For instance, in three dimensions it is possible to consider higher spin gauge theory in flat space-times [63][64][65][66][67], unlike the situation in higher dimensions where a non-zero cosmological constant is required. 3 Moreover, in three dimensions higher spin gauge theories with only a finite number of higher spin fields can be constructed [71]. In the relativistic case, such theories assume the form of Chern-Simons theories, for a gauge group that is a suitable finite-dimensional extension of the three-dimensional (A)dS and Poincaré groups. For theories with integer spins ranging from 2 to N in AdS space-time, this gauge group is given by SL(N, R) ⊗ SL(N, R). Here, we will restrict ourselves for simplicity to 'spin-3 theory' for which N = 3, although our analysis can straightforwardly be generalized to arbitrary N .
In this paper, we will thus extend the discussion of kinematical algebras of [1] to theories in three space-time dimensions that include a spin-3 field coupled to gravity. In particular,

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and can therefore be read independently. Finally, in 5 we end with our conclusions and an outlook for future work.

Kinematical spin-3 algebras
In this section, we will be concerned with three-dimensional kinematical spin-3 algebras, i.e. generalized space-time symmetry algebras of theories of interacting, massless spin-2 and spin-3 fields. In particular, following Bacry and Lévy-Leblond [1] we will classify all such algebras that can be obtained by combining different Inönü-Wigner (IW) contraction [81] procedures from the algebras that underlie spin-3 gravity in AdS 3 and dS 3 . After recalling the latter, we will present all possible ways of contracting them, such that non-trivial kinematical spin-3 algebras are obtained, via a classification theorem. The proof of this theorem is relegated to appendix B. Combining different of these contraction procedures leads to various kinematical spin-3 algebras, some of which will be discussed in the next section as a starting point for considering Carroll and Galilei spin-3 gravity Chern-Simons theories.

AdS 3 and dS 3 spin-3 algebras
Spin-3 gravity in (A)dS 3 [79,80] can be written as a Chern-Simons theory for the Lie algebra sl(3, R) ⊕ sl(3, R) (where the ⊕ denotes the direct sum as Lie algebras) for AdS 3 or sl(3, C) (viewed as a real Lie algebra) for dS 3 . In the following we will often denote the higher spin algebra sl(3, R) ⊕ sl(3, R), realizing Spin-3 gravity in AdS 3 , by hs 3 AdS. Similarly, we indicate the higher spin algebra sl(3, C), realizing Spin-3 gravity in dS 3 , by hs 3 dS. In both cases, the algebra consists of the generators of Lorentz transformationsĴ A and translationsP A along with 'spin-3 rotations'Ĵ AB and 'spin-3 translations'P AB , that are traceless-symmetric in the (AB) indices (A = 0, 1, 2): 5 J AB =Ĵ BA , η ABĴ AB = 0 , P AB =P BA , η ABP AB = 0 . (2.1) Here, η AB is the three-dimensional Minkowski metric. We will often refer to {Ĵ A ,P A } as the 'spin-2 generators' or the 'spin-2 part' and similarly to {Ĵ AB ,P AB } as the 'spin-3 generators' or 'spin-3 part'. Their commutation relations are given by [79,80] Ĵ A ,Ĵ B = ABCĴ C , Ĵ A ,P B = ABCP C , where the upper sign refers to hs 3 AdS and the lower sign to hs 3 dS. Note that the first two lines constitute the isometry algebra of (A)dS 3 , i.e. sl(2, R) ⊕ sl(2, R) for AdS 3 and sl(2, C), viewed as a real Lie algebra, for dS 3 .

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For future reference, we mention that the above algebras are equipped with a nondegenerate ad-invariant symmetric bilinear form that we will denote by · , · (where the · are placeholders for generators) and that we will henceforth call the 'invariant metric'. Its non-zero components are given by Note that this represents an invariant metric for both hs 3 AdS and hs 3 dS. The existence of this metric allows one to construct Chern-Simons actions for the algebras hs 3 AdS and hs 3 dS, that correspond to the actions for spin-3 gravity in (A)dS 3 [79,80]. In the following, it will prove convenient to introduce a time-space splitting of the indices A = {0, a; a = 1, 2}. We will thereby use the following notation: Note that we have left out the generatorsP 00 andĜ 00 here. These generators are not independent, due to the tracelessness constraint (2.1) and in the following we will eliminate them in favour of P ab and G ab . After these substitutions, the commutation relations of hs 3 (A)dS in this new basis are given in the first column of table 6.

All kinematical spin-3 algebras by contracting hs 3 (A)dS
Before discussing spin-3, it is convenient to start with giving a short review of the spin-2 case [1]. Since both the spin-2 and spin-3 cases make use of the IW contraction we first discuss this procedure. We will use this as an opportunity to introduce some terminology that will be used throughout this paper. Starting from a Lie algebra g, one can choose a subalgebra h and consider the decomposition g = h + i where + denotes the direct sum as vector spaces (not as Lie algebras). Upon rescaling the generators of i with a so-called contraction parameter , i → i, the commutation relations of g assume the following form One thus sees that the limit → ∞ is well-defined. 6 Taking this limit leads to an inequivalent algebra, that is a semidirect sum i h, for which i is an abelian ideal This algebra is called the IW contraction of g with respect to h. The IW contraction is called 'trivial' if either h = g or i = g. The procedure that leads to an IW contraction, i.e. that consists of choosing a subalgebra h, rescaling the generators of i as i → i and taking → ∞, will be denoted as the 'IW contraction procedure' in this paper. Note that a nontrivial IW contraction procedure is uniquely specified by a suitable choice of the subalgebra h ⊂ g. Not all possible subalgebras, however, lead to interesting IW contractions that can e.g. be interpreted as kinematical algebras. For spin-2, the question which contraction procedures of the isometry algebras of AdS or dS lead to kinematical algebras, has been addressed by Bacry and Lévy-Leblond [1]. In particular, they have shown that there are only four different IW contraction procedures of the AdS or dS algebras that lead to kinematical algebras. These have been called 'space-time', 'speed-space', 'speedtime' and 'general' in [1]. Effectively, the first three of these contractions can be described by either taking a limit of the (A)dS radius or the speed of light c. Specifically, the space-time contraction corresponds to → ∞, the speed-time contraction corresponds to c → 0 and the speed-space contraction corresponds to c → ∞. However, in this work we suppress factors of and c. The general contraction procedure can also be obtained as a sequential contraction of the other three and therefore does not provide us with a new algebra. Moreover, it has been shown that there are in total 8 possible kinematical algebras 7 that can be obtained by combining different IW contraction procedures of the AdS or dS isometry algebras. We have summarized the four IW contraction procedures in table 1, by indicating the subalgebra h with respect to which the contraction procedure is taken, as well as the generators that form the abelian ideal i. The names of the eight kinematical algebras of [1], along with the symbols we will use to denote them, are given in table 2. The IW contraction procedures and the contracted algebras that we discussed so far can be conveniently summarized as a cube, see figure 1.

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We next consider the spin-3 case where, following the spin-2 case, we will obtain a classification of all possible contraction procedures 8 of hs 3 AdS and hs 3 dS by listing all their possible subalgebras. We start from hs 3 (A)dS since these are semisimple algebras and can 7 The possible kinematical algebras considered in [1] are all possible space-time symmetry algebras that obey the assumptions that space is isotropic and therefore their generators have the correct (H is a scalar, P, J, G are vectors) transformation behavior under rotations. Furthermore, parity and time-reversal are automorphisms and boosts are non-compact. 8 Here, we will classify different contraction procedures, in the sense defined above as different choices of subalgebra h. This does not mean that all these contraction procedures lead to non-isomorphic Lie algebras. Indeed, in the analysis of [1] e.g. one can see that the space-time and speed-time contraction procedures applied to the AdS3 isometry algebra lead to two contractions that are both isomorphic to the Poincaré algebra. We should however mention that these algebras are isomorphic in the mathematical sense; physically they can be regarded as non-equivalent as the isomorphism that relates them corresponds to an interchange of boost and translation generators. Note also that the different contraction procedures that are classified here are not necessarily independent. As an example, one can check that the general contraction procedure of   . This cube summarizes the sequential contractions starting from (A)dS. The lines represent contraction procedures and the dots represent the resulting contractions. We consider contraction procedures starting from AdS and dS simultaneously. Each dot can therefore represent one contraction, if the contraction procedures from AdS and dS lead to the same algebra, or two contractions, if the contraction procedures from AdS and dS lead to two different results. We have indicated this in the cube by using single lines, for contraction procedures that lead to the same contraction, and double lines otherwise. Dashed lines have no specific meaning except that they should convey the feeling of a three-dimensional cube. therefore not be viewed as contractions themselves (as nontrivial contraction procedures always lead to algebras with an abelian ideal that are thus not semisimple). Now, in order to obtain contractions that can be identified as interesting kinematical spin-3 algebras, we will impose two restrictions:

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• When restricted to the spin-2 part of the algebra, the contraction procedures should correspond to those considered in table 1. This ensures that the spin-2 parts of the algebras obtained by various combinations of these contraction procedures correspond to the kinematical algebras of [1].
• Furthermore, we will also demand that in the resulting contraction not all commutators of the spin-3 part are vanishing. This requirement is motivated by the fact that we are interested in using these contractions to describe fully interacting theories of massless spin-2 and spin-3 fields. Indeed, as we will show later on, for some of the algebras obtained here, one can construct a Chern-Simons action for spin-2 and spin-3 fields. Only when the commutators of the spin-3 part are not all vanishing, do the spin-3 fields contribute to the equations of motion of the spin-2 fields.
All ways of contracting hs 3 AdS and hs 3 dS that obey these two restrictions can then be summarized by the following theorem: Theorem 1. All possible IW contraction procedures, that reduce to those considered in table 1 when restricted to the spin-2 part and that are non-abelian on the subspace spanned by the spin-3 generators {J a , H a , G ab , P ab }, are given by 10 'democratic' contraction procedures that are specified in  to be supplemented with spin-3 generators, in order to have a contraction with a nonabelian spin-3 part. The proof then proceeds by enumerating, for each of the contraction procedures of table 1, all possibilities in which spin-3 generators can be added to h such that one still obtains a subalgebra, that leads to a contraction with a non-abelian spin-3 part. We refer to appendix C for the explicit Lie algebras of the contraction procedures given in table 3.
Finally, let us comment on the terminology 'democratic' and 'traceless'. This terminology stems from the fact that the three independent generators contained in P ab (G ab ) form a real, reducible representation of J, that can be split into a tracefree symmetric part consisting of the generators {P 12 , P 22 − P 11 } ({G 12 , G 22 − G 11 }) and a trace part P 11 + P 11 (G 11 +G 22 ). The democratic contraction procedures are such that the subalgebra h contains both tracefree symmetric and trace components of P ab (G ab ), if present. In some cases, it is not necessary to include the trace component in h in order to obtain a valid subalgebra. This is the case for the democratic contraction procedures, numbered 4, 8 and 10 in table 3. Moving the trace component from h to i leads to the traceless cases 4a, 4b, 4c, 8a and 10a in table 4. In the last two remaining cases both the tracefree symmetric part of G ab (P ab ) and the trace part of P ab (G ab ) belong to the subalgebra h. Doing this leads to the traceless cases 8b and 10b.
The democratic contractions can again be summarized as a cube, see figure 2.

Spin-Carroll, Galilei and extended Bargmann Chern-Simons theories
In the previous section, we have classified all possible contraction procedures of the spin-3 AdS 3 and dS 3 algebras. Combining some of these contraction procedures can lead to algebras whose spin-2 part corresponds to the Carroll or Galilei algebra. Here, we will study these cases in more detail. In particular, we will be concerned with constructing Chern-Simons theories for these spin-3 algebras, or suitable extensions thereof. This extends [75] where the case of spin-2 Carroll and spin-2 Galilei gravity is discussed.
Assuming that there is an invariant metric · , · available, the Chern-Simons action can be constructed as where A is an algebra-valued gauge field. Using the invariant metric (2.3), one can find the Chern-Simons actions for spin-3 gravity in (A)dS 3 [79,80]. Performing the first contraction procedure of table 3, i.e. rescalingP A → P A ,P AB → P AB and taking the limit → ∞, leads to the spin-3 Poincaré algebra hs 3 poi1 of table 6. This algebra can be equipped with the same invariant metric (2.3) and the associated Chern-Simons action leads to spin-3 gravity in three-dimensional flat space-time [63,64]. In order to construct Chern-Simons actions for Carroll and Galilei spin-3 algebras, one therefore needs to know whether these algebras can be equipped with an invariant metric. In this respect, it is useful to note that it is not always true that the IW contraction of an algebra equipped with an invariant metric, also admits one. A counter-example is provided by the three-dimensional Carroll and Galilei algebras which both arise as IW contractions of the Poincaré algebra, that in three dimensions has an invariant metric. However, only the Carroll algebra admits an invariant metric; the Galilei algebra does not. Naively, one can thus not construct a Chern-Simons action for the Galilei algebra. Fortunately, there exists JHEP01(2017)114 an extension of the Galilei algebra, the so-called Extended Bargmann algebra [13,[72][73][74], that can be equipped with an invariant metric and for which a Chern-Simons action can be constructed.
In this section, we will show that similar results hold in the spin-3 case. In particular, we will see that the spin-3 versions of the Carroll algebra admit an invariant metric and that a Chern-Simons action can be straightforwardly constructed. The spin-3 versions of the Galilei algebra do not have an invariant metric but a theorem due to Medina and Revoy [82] implies that they can be extended to algebras with an invariant metric. We will then explicitly construct these 'spin-3 Extended Bargmann' algebras and their associated Chern-Simons actions. In this way, we will obtain spin-3 versions of Carroll gravity [30,75] and Extended Bargmann gravity [13,[72][73][74].
We will first treat the case of spin-3 Carroll gravity in section 3.1, while the spin-3 Extended Bargmann gravity case will be discussed in section 3.2. In both cases, we will also study the equations of motion, at the linearized level. This will allow us to interpret the Chern-Simons actions for these theories as suitable spin-3 generalizations of the actions of Carroll and Extended Bargmann gravity, in a first order formulation. In particular, this linearized analysis will show that some of the gauge fields appearing in these actions can be interpreted as generalized vielbeine, while others can be viewed as generalized spin connections. The latter in particular appear only algebraically in the equations of motion and are therefore dependent fields that can be expressed in terms of other fields. We will give these expressions. In some cases, we will see that not all spin connection components become dependent. We will argue that the remaining independent spin connection components can be viewed as Lagrange multipliers that implement certain constraints on the geometry. For simplicity, we will restrict ourselves to Carroll and Galilei spin-3 gravity theories. The analysis provided here can be straightforwardly extended to include a cosmological constant.

Spin-3 Carroll gravity
There are four distinct ways of contracting hs 3 (A)dS, such that a spin-3 algebra whose spin-2 part coincides with the Carroll algebra is obtained. These four ways correspond to combining the contraction procedures 1 and 5, 1 and 6, 2 and 5 or 2 and 6 of table 3 respectively. We will denote the resulting algebras as hs 3 car1, hs 3 car2, hs 3 car3 and hs 3 car4. Their structure constants are summarized in table 9. Note that hs 3 car3 and hs 3 car4 each come in two versions, since we apply the IW contraction procedures to AdS and dS simultaneously. These versions differ in the signs of some of their structure constants, as can be seen from table 9. The existence of these different versions when applying the contraction procedures 2 and 5 (or 2 and 6) stems from the fact that the combination of these contraction procedures leads to different algebras, depending on whether one starts from hs 3 AdS or from hs 3 dS. By contrast, applying contraction procedures 1 and 5 (or 1 and 6) on hs 3 AdS and hs 3 dS leads to the same result, namely hs 3 car1 (or hs 3 car2).
By examining the structure constants of table 9, one can see that hs 3 car1 (hs 3 car2) and hs 3 car3 (hs 3 car4) are related via the following interchange of generators

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plus potentially some sign changes in structure constants, as mentioned in the previous paragraph. The structure of the Chern-Simons theories will therefore be very similar for hs 3 car1 (hs 3 car2) and hs 3 car3 (hs 3 car4). In the following, we will therefore restrict ourselves to hs 3 car1 and hs 3 car2. We will now discuss the Chern-Simons theories for these two cases in turn.

Chern-Simons theory for hs 3 car1
The commutation relations of hs 3 car1 are summarized in the first column of table 9. This algebra admits the following invariant metric This can be obtained either by direct computation or by applying the contraction procedures 1 and 5 of table 3 on the invariant metric (2.3) of hs 3 (A)dS (suitably rescaled with the contraction parameters). Using the commutation relations of hs 3 car1 and the invariant metric (3.3), the Chern-Simons action (3.1) and its equations of motion can be explicitly written down. Here, we will be interested in studying the action and equations of motion, linearized around a flat background solution 9 given bȳ We will therefore assume that the gauge field is given by this background solutionĀ µ , plus fluctuations around this background Here, τ µ can be interpreted as a linearized time-like vielbein, e µ a as a linearized spatial vielbein, while ω µ and B µ a can be viewed as linearized spin connections for spatial rotations and boosts respectively. Similarly, τ µ a , e µ ab , ω µ a and B µ ab can be interpreted as spin-3 versions of these linearized vielbeine and spin connections. Using the expansion (3.5) in the Chern-Simons action (3.1) and keeping only the terms quadratic in the fluctuations, one finds the following linearized action: For fields in this flat background solution, the curved µ index becomes equivalent to a flat one. In the following, we will therefore denote the time-like and spatial values of the µ index by 0 and a. The a index can moreover be freely raised and lowered using a Kronecker delta. We will often raise or lower spatial a indices on field components (even if it leads to equations with non-matching index positions on the leftand right-hand-sides), to make more clear which field components are being meant.

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The linearized equations of motion corresponding to this action are given by The equations contain the spin connections ω µ , B µ a , ω µ a and B µ ab only in an algebraic way. These equations can thus be solved to yield expressions for some of the spin connection components in terms of the vielbeine and their derivatives. Let us first see how this works for the spin-2 spin connections ω µ and B µ a . The equation R 0a (H) = 0 can be straightforwardly solved for B 0 a : Similarly, the equation R ab (H) = 0 (or equivalently ab R ab (H) = 0) can be solved for B c c (the spatial trace of B µ a ): From R ab (P c ) = 0 (or equivalently ab R ab (P c ) = 0) one finds the spatial part of ω µ : Finally, let us consider the equation R 0a (P b ) = 0. The anti-symmetric part of this equation ab R 0a (P b ) = 0 can be solved for the time-like part of ω µ : The symmetric part R 0(a (P b) ) = 0 does not contain any spin connection and can be viewed as a constraint on the geometry

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In summary, we find that R µν (H) = 0 and R µν (P a ) = 0 lead to the constraint (3.13) as well as the following solutions for ω µ and B µ a whereB b a is an undetermined traceless tensor. The boost connection B µ a is thus not fully determined in terms of τ µ and e µ a .
A similar reasoning allows one to solve for certain components of the spin-3 connections ω µ a and B µ ab . In particular, the equation R ab (H c ) = 0 can be solved for B d da , a spatial trace of B µ ab : The equation R ab (P cd ) = 0 can be solved for the symmetric, spatial part of ω µ a : The anti-symmetric, spatial part of ω µ a can be found from R 0a (H a ) = 0: From the other equations contained in R 0b (H a ) = 0 one then finds .
(3.18) The equation R 0a (P bc ) = 0 can be divided into a part that is fully symmetric in the indices a, b, c and a part that is of mixed symmetry: The equation R MS 0a (P bc ) = 0 can be solved for ω 0 a , by noting that The solution one finds is given by

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The fully symmetric part R S 0a (P bc ) = 0 can not be used to solve for other spin connection components. Rather, it should be viewed as a constraint on the geometry: This constraint can be slightly simplified. By contracting it with δ bc , one finds that Using this, one finds that (3.23) simplifies to One thus finds for the spin-3 sector, that the equations R µν (H a ) = 0 and R µν (P ab ) = 0 lead to the constraint (3.25) and the following solutions for ω µ a and B µ ab : whereB c ab is an arbitrary tensor obeyingB b ba = 0. As for the spin-2 sector, one thus finds that the spin-3 boost connection B µ ab can not be fully determined in terms of τ µ a and e µ ab .
It is interesting to see what role the undetermined componentsB b a andB c ab play. In particular, one can check how these components appear in the Lagrangian and what their equations of motion are. Upon partial integration in the action (3.6), one finds that the terms in the Lagrangian involving B µ a can be written as The traceless spatial componentsB b a of B ρ a thus couple to (3.28)

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This can however be rewritten as One thus sees thatB b a acts as a Lagrange multiplier for R 0(a (P b) ) = 0, which led to the constraint (3.13). Similarly, one can check thatB c ab plays the role of Lagrange multiplier for the constraint (3.25).

Chern-Simons theory for hs 3 car2
The commutation relations of hs 3 car2 can be obtained by successively applying the contraction procedures 1 and 6 of  4) is a solution of the full non-linear equations of motion of the Chern-Simons theory for hs 3 car2. As before, we will study the Chern-Simons action and its equations of motion, linearized around this background. Adopting the linearization ansatz (3.5), the linearized Chern-Simons action is found to be given by The linearized equations of motion derived from this action are As in the case of hs 3 car1, the equations R µν (H) = 0, R µν (P a ) = 0, R µν (H a ) = 0 and R µν (P ab ) = 0 contain the spin connections ω µ , B µ a , ω µ a and B µ ab algebraically. These equations can thus be used to express some spin connection components in terms of the vielbeine. Since the linearized equations of motion for hs 3 car1 and hs 3 car2 coincide for JHEP01(2017)114 the spin-2 sector, the solutions for ω µ and B µ a are again given by (3.14). As before, the traceless spatial partB b a of B µ a acts as a Lagrange multiplier for the constraint (3.13).
In the spin-3 sector, one can solve the equations R cd (P ab ) = 0 for the symmetric, spatial part of ω µ a : while the remaining equations in R 0c (P aa ) = 0, together with R bc (H a ) = 0 lead to The traceless part of R 0b (H a ) = 0 can be used to solve for B 0 ab : One thus finds whereω is undetermined. By examining the action (3.30), one can see thatω plays the role of a Lagrange multiplier for the constraint (3.36).

Spin-3 Galilei and extended Bargmann gravity
In the previous section, we have studied Carroll spin-3 algebras, whose spin-2 part corresponds to the Carroll algebra. Using the contraction procedures of table 3, one can also obtain non-relativistic spin-3 algebras, that contain the Galilei algebra. As in the Carroll

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case, there are four distinct ways of doing this, namely by successively applying the contraction procedures 1 and 3, 1 and 4, 2 and 3 or 2 and 4 of table 3. We have called the resulting algebras hs 3 gal1, hs 3 gal2, hs 3 gal3 and hs 3 gal4 respectively and summarized their commutation relations in table 10. As in the Carroll case, hs 3 gal3 and hs 3 gal4 each come in two different versions, depending on whether one applies the combination of contraction procedures on hs 3 AdS or hs 3 dS. They are again structurally similar to hs 3 gal1 and hs 3 gal2. We will therefore restrict our discussion here to these two cases.
In contrast to the spin-3 Carroll algebras, whose invariant metrics arose from applying the relevant contraction procedures on (2.3), a similar reasoning for the spin-3 Galilei algebras leads to degenerate bilinear forms. One can in fact show by direct computation that they can not be equipped with a nondegenerate symmetric invariant bilinear form. 10 It could be interesting to investigate these algebras, given explicitly in table 10, and their degenerate bilinear forms. For the spin-2 case, this has been done in [75]. Due to the degeneracy of the bilinear form, some of the fields appear without kinetic term in the action (see equation (3.1)) and are therefore not dynamical. In the spin-2 case, one can nevertheless interpret these non-dynamical fields as Lagrange multipliers for geometrical constraints, similarly to what happens in the Carroll cases of the previous section. Although it would be interesting to see whether similar results hold for the higher spin case, we will not do this here and instead we will look at Chern-Simons theories where each field has a kinetic term. These can not be based on the spin-3 Galilei algebras, but interestingly, a theorem due to Medina and Revoy [82] implies that these algebras can be extended to algebras that admit an invariant metric, i.e. a nondegenerate ad-invariant symmetric bilinear form. Remarkably, in this way one ends up with a spin-3 version of the Extended Bargmann algebra. This procedure can be stated as follows [82,83].
Theorem 2. Consider a Lie algebra that is a semi-direct sum of Lie algebras g (with generators G i ) and h (with generators H α ), whose commutation relation are given by Suppose furthermore that g is equipped with an invariant metric · , · g G i , G j g = g ij , (3.39) that is left invariant by the action of h on g, given by the second commutation relation of (3.38). Denote the dual of h by h * (with generators H α ). Then, g + h + h * (where + denotes a direct sum as vector spaces) forms a Lie algebra d with commutation relations

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where f ijα = f αi k g kj . This Lie algebra d, called the double extension of g by h, moreover admits an invariant metric · , · d whose non-zero components are given by where h αβ represents a possibly degenerate symmetric invariant bilinear form on h.
This theorem can be applied to the ordinary Galilei algebra in three dimensions and yields the so-called Extended Bargmann algebra [13,[72][73][74], that extends the Galilei algebra with two central extensions. Applying the theorem to hs 3 gal1 and hs 3 gal2 yields two spin-3 algebras, that we will denote, in hindsight, by hs 3 ebarg1 and hs 3 ebarg2 (since they have an Extended Bargmann spin-2 subalgebra).
The algebra hs 3 ebarg1 can be obtained by applying the Medina-Revoy theorem to hs 3 gal1. Indeed, with the choices g = {P a , G a , P ab , G ab }, h = {H, J, H a , J a } and the assumptions of the theorem are fulfilled and the algebra hs 3 ebarg1 can be constructed.
Denoting the generators of h * by {H * , J * , H * a , J * a }, the commutation relations of hs 3 ebarg1 are given in table 5. The invariant metric of hs 3 ebarg1 is explicitly given by Similarly, starting from hs 3 gal2 and using the Medina-Revoy theorem with the choices g = {P a , G a , H a , J a }, h = {H, J, P ab , G ab } and the algebra hs 3 ebarg2 can be constructed. Denoting the generators of h * by {H * , J * , P * ab , G * ab }, its commutation relations are given in table 5. This algebra admits the following invariant metric Note that for both hs 3 ebarg1 and hs 3 ebarg2 the generators {H, J, P a , G a , H * , J * } form a subalgebra that coincides with the Extended Bargmann algebra. The Chern-Simons theories based on these algebras can therefore be viewed as spin-3 extensions of Extended Bargmann gravity, studied in [13,[72][73][74]. In the next subsections, we will discuss these spin-3 Extended Bargmann gravity theories, at the linearized level.

Chern-Simons theory for hs 3 ebarg1
Using the structure constants of table 5 and the invariant metric (3.43), the Chern-Simons action for hs 3 ebarg1 can be constructed. As before, we will restrict ourselves to a linearized analysis. In particular, one can show that the followinḡ is again a solution of the full non-linear equations of motion. Similar in spirit to (3.5), we consider the following linearization ansatz The linearized field equations can be obtained by putting the following linearized field strengths to zero As in the Carroll cases, the spin connections ω µ , B µ a , ω µ a and B µ ab appear algebraically in the equations

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and one can use these equations to express the spin connections in terms of the other fields.
In the spin-2 sector, one straightforwardly finds that R µν (P a ) = 0 and R µν (J * ) = 0 lead to Note that all components of ω µ and B µ a are now uniquely determined. This follows from the fact that the system of equations R µν (P a ) = 0 and R µν (J * ) = 0 contains as many independent equations as there are independent components of ω µ and B µ a .
In the spin-3 sector, one can use the equations to obtain expressions for the connections ω µ a and B µ ab . Unlike for the spin-2 sector, the system of equations (3.51) is overdetermined, i.e. the number of equations exceeds the number of independent components of ω µ a and B µ ab . One can choose some of the equations of (3.51) to express ω µ a and B µ ab in terms of τ µ a , e µ ab , ω µ a * and their derivatives.
The remaining equations can then be interpreted as geometrical constraints, once the expressions for the spin connections are used. Dividing the system (3.51) into equations that are solved for connections and equations that remain as constraints, can not be done in a unique way. One convenient way is to use to solve for ω µ a and R 0b (J a * ) − 1 2 δ a b R 0c (J c * ) = 0 , R bc (J a * ) = 0 , R 0c (P ab ) − 1 2 δ ab R 0c (P dd ) = 0 , (3.53) to solve for B µ ab , while the remaining equations can be interpreted as geometrical constraints. The solutions for the connections are then explicitly given by

Chern-Simons theory for hs 3 ebarg2
The Chern-Simons theory for hs 3 ebarg2, with structure constants given in table 5   These equations are the same as in the case of hs 3 ebarg1 and one finds the same solution (3.50). For the spin-3 sector, one has to solve the equations As in the hs 3 ebarg1 case, this is an overdetermined set of equations, that can be used to obtain expressions for ω µ a and B µ ab along with geometrical constraints. One convenient way of doing this, is by using the equations to solve for ω µ a . An expression for B µ ab can be found as a solution of

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The connections ω µ a and B µ ab are then given by

Carroll gravity as example
In this section we address whether there are interesting infinite extensions of the algebras discussed above, in the same way that the global conformal algebra in two dimensions gets extended to the Virasoro algebra by imposing Brown-Henneaux boundary conditions [76].
Rather than being as comprehensive as in the other sections we focus here on a specific simple example. In fact, we drop the higher spin fields and consider spin-2 Carroll gravity, defined by a Chern-Simons gauge theory with action (3.1) where the connection 1-form takes values in the spin-2 Carroll algebra (a = 1, 2), whose non-vanishing commutation relations read where we use the convention 12 = +1 for the antisymmetric -symbol. The invariant metric has the non-vanishing entries fully compatible with (3.3) in the absence of higher spin generators. Our main goal is not just to find some infinite extension of the algebra (4.2) (this always exists at least in the form of the loop algebra of the underlying gauge algebra, see e.g. [84]; for AdS 3 gravity such boundary conditions were investigated recently in [85]), but rather to find an extension that has a 'nice' geometric interpretation along the lines of the Brown-Henneaux boundary conditions. This means that we want to achieve a suitable Drinfeld-Sokolov type of reduction where not all algebraic components of the connection are allowed to fluctuate. The words 'nice' and 'suitable' here mean that, in particular, we want that the appropriate Carroll background geometry as part of our spectrum of physical

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states is allowed by our boundary conditions, and that all additional states are fluctuations around this background. First, we recall some basic aspects of Carroll geometry.
The Carroll-zweibein for the flat background geometry in some Fefferman-Graham like coordinates should take the form so that the corresponding two-dimensional line-element reads We shall refer to ρ as 'radial coordinate' and to ϕ as 'angular coordinate', assuming ϕ ∼ ϕ + 2π. Moreover, on the background the time-component should be fixed as We proceed now by stating the result for the boundary conditions that define our example of Carroll gravity and discuss afterwards the rationale behind our choices as well as the consistency of the boundary conditions by proving the finiteness, integrability, nontriviality and conservation of the canonical boundary charges. We follow the general recipe reviewed e.g. in [57,86]. First, we bring the connection (4.1) into a convenient gauge (see for instance [87]) where the group element b(ρ) = e ρP 2 (4.8) is fixed as part of the specification of our boundary conditions, δb = 0. The boundary connection a does not depend on the radial coordinate ρ and is given by where µ is arbitrary but fixed, δµ = 0, while all other functions are arbitrary and can vary. This means that the allowed variations of the boundary connection are given by The full connection in terms of the boundary connection is then given by and acquires its non-trivial radial dependence through the last term, ρ [a, P 2 ] = ρ (P 1 − g 1 (t, ϕ) H) dϕ. Only the ϕ-component of the connection is then allowed to vary.

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The above boundary conditions lead to Carroll-geometries of the form ds 2 (2) = ρ + p 1 (t, ϕ) 2 + p 2 (t, ϕ) 2 dϕ 2 + 2p 2 (t, ϕ) dϕ dρ + dρ 2 (4.13) and τ = µ(t, ϕ) dt + h(t, ϕ) − ρ g 1 (t, ϕ) dϕ . (4.14) Thus, we see that to leading order in ρ the background line-element (4.5) is recovered from (4.13), plus subleading (state-dependent) fluctuations captured by the functions p a (t, ϕ). As we shall see in the next paragraph the functions p a and g a are t-independent on-shell. In the metric-formulation our boundary conditions can be phrased as Note that while the asymptotic form of the two-dimensional line-element (4.15) may have been guessed easily, the specific form of the time-component (4.16) is much harder to guess, particularly the existence of a 'shift'-component proportional to dϕ that grows linearly in ρ.
Fortunately, the Chern-Simons formulation together with the gauge choice (4.7) minimizes the amount of guesswork needed to come up with meaningful boundary conditions. We consider now the impact of the equations of motion on the free functions in the boundary connection (4.9). Gauge-flatness F = 0 implies As a consequence, we get the on-shell conditions (which also could be called 'holographic Ward identities') ∂ t p a = ∂ t g a = 0 ∂ t h = ∂ ϕ µ .
where O(δa ϕ ) denotes all the allowed variations displayed in (4.10). It is useful to decompose λ with respect to the algebra (4.1). (4.20) The first line in (4.19) establishes the time-independence of λ, while the second line yields the consistency condition ∂ ϕ λ J = 0 (4.21)

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as well as the transformations rules Applying the Regge-Teitelboim approach [88] to Chern-Simons theories yields the following background-independent result for the variation of the canonical boundary charges which in our case expands to The canonical boundary charges are manifestly finite since the ρ-dependence drops out in (4.23); they are also integrable in field-space since our λ is state-independent.
The result (4.25) clearly is non-trivial in general. To conclude the proof that we have meaningful boundary conditions we finally check conservation in time, using the on-shell relations (4.18) as well as the time-independence of λ, see (4.19a): By virtue of (4.21) we see that the last integrand vanishes and thus we have established charge conservation on-shell: Since our canonical boundary charges (4.25) are finite, integrable in field space, non-trivial and conserved in time the boundary conditions (4.7)-(4.12) are consistent and lead to a non-trivial theory. For later purposes, it is useful to note that due to the constancy of λ J only the zero mode charge associated with the function h can be non-trivial. This means that we can gauge-fix our connection using proper gauge transformations such that h = const. We now introduce Fourier modes in order to be able to present the asymptotic symmetry algebra in a convenient form. 11 P a n := 1 2π dϕ e inϕ g a (t, ϕ) EOM , (4.28a) G a n := 1 2π dϕ e inϕ p a (t, ϕ) EOM , (4.28b) (4.28c) 11 There is no meaning to the index positions in this section. The only reason why we write P a n and G a n instead of corresponding quantities with lower indices is that our current convention is easier to read.

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A few explanations are in order. Due to our off-diagonal bilinear form (4.3) we associate the n th Fourier mode of the functions g a (p a ) with the generator P a n (G a n ). For the same reason we associate J with minus the zero mode of h. Finally, the subscript 'EOM' means that all integrals are evaluated on-shell, in which case all t-dependence drops out (and in the last integral also all ϕ-dependence).
We make a similar Fourier decomposition of the gauge parameters λ i , where i refers to the generators P a , G a and J; the parameter λ H is not needed since it does not appear in the canonical boundary charges (4.25), so all gauge transformations associated with it are proper ones and can be used to make h constant.  δP a n = −inλ Ga From the variations (4.30) we can read off the asymptotic symmetry algebra, using the fact that the canonical generators generate gauge transformations via the Dirac bracket Converting Dirac brackets into commutators then establishes the asymptotic symmetry algebra as the commutator algebra of the infinite set of generators P a n , G a n and J. The central element of this algebra will be associated with (minus) H, concurrent with the notation of (4.2). Evaluating (4.30) yields 12 [J, P a n ] = ab P b n , (4.31a) [J, G a n ] = ab G b n , (4.31b) [P a n , where all commutators not displayed vanish. We have thus succeeded in providing an infinite lift of the Carroll algebra (4.2), which is contained as a subalgebra of our asymptotic symmetry algebra (4.31) by restricting to the zero-mode generators P a = P a 0 , G a = G a 0 in addition to J and H. As a simple consistency check one may verify that the Jacobi identities indeed hold. The only non-trivial one to be checked is the identity [[J, P a n ], G b m ] + cycl. = 0.

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We conclude this section with a couple of remarks. The boundary conditions (4.7)-(4.9) by no means are unique and can be either generalized or specialized to looser or stricter ones, respectively. In particular, we have switched off nearly all 'chemical potentials' in our specification of the time-component of the connection (4.9b), and it could be of interest to allow arbitrary chemical potentials. Apart from this issue there is only one substantial generalization of our boundary conditions, namely to allow for a state-dependent function in front of the generator J in the angular component of the connection (4.9a). As mentioned in the opening paragraph of this section, in that case the expected asymptotic symmetry algebra is the loop algebra of the Carroll algebra (4.2). In principle, it is possible to make our boundary conditions stricter, but that would potentially eliminate interesting physical states like some of the Carroll geometries (4.13), (4.14). 13 Thus, while our choice (4.7)-(4.9) is not unique it may provide the most interesting set of boundary conditions for spin-2 Carroll gravity. Using the same techniques it should be straightforward to extend the discussion of this section to higher spin Carroll gravity and related theories discussed in this paper.

Discussion
In this paper, we have extended the work of Bacry and Lévy-Leblond [1] by classifying all possible kinematical algebras of three-dimensional theories of a spin-3 field coupled to gravity, that can be obtained via (sequential) Inönü-Wigner contraction procedures of the algebras of spin-3 gravity in (A)dS. This classification can be found in section 2 and the resulting possible kinematical algebras, along with their origin via contraction, are summarized in figure 2. We have summarized the commutation relations of the algebras in tables 6-13. The algebras of tables 9 and 10 are suitable generalizations of the Carroll and Galilei algebras, that correspond to the ultra-relativistic and non-relativistic limits of the Poincaré algebra. These algebras have been used in section 3 as a starting point to construct higher spin generalizations of ultra-relativistic and non-relativistic gravity theories. We have argued that one can easily construct a Chern-Simons action for the spin-3 Carroll algebras, that leads to a spin-3 generalization of Carroll gravity. We have moreover shown that Chern-Simons actions can be written down for suitable extensions of the spin-3 Galilei algebras, that lead to spin-3 generalizations of Extended Bargmann gravity. The algebras constructed in this paper are finite-dimensional. We have shown in section 4 that the three-dimensional Carroll algebra admits an infinite-dimensional extension, that is the asymptotic symmetry algebra of Carroll gravity with suitable boundary conditions. This can be taken as a hint that similar results hold for the non-and ultra-relativistic algebras constructed in this paper, as well as for the spin-2 algebras whose infinite-dimensional extensions have not been addressed in the literature yet.
There are several questions that are worthwhile for future study. The non-and ultrarelativistic spin-3 gravity theories constructed here, are given in the Chern-Simons (i.e. first

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order 'zuvielbein') formulation. It is interesting to see whether a metric-like formulation can be constructed and whether the linearized field equations can be rewritten as Fronsdallike equations. The results for the linearized spin connections given in section 3 should be useful in this regard.
In this paper, we have restricted ourselves to spin-3 theories, by considering algebras that are obtained via (sequential) Inönü-Wigner contraction procedures of sl(3, R)⊕sl(3, R) or sl (3, C). This analysis can be extended to theories with fields up to spin N , by considering contraction procedures of sl(N, R) ⊕ sl(N, R) or sl(N, C) [89]. One can then study the non-and ultra-relativistic gravity theories that arise in this way and in particular investigate the types of boundary conditions that lead to interesting asymptotic symmetry algebras. It would be particularly interesting to see whether it is possible to construct nonand ultra-relativistic versions of non-linear W -algebras.
Another research direction concerns the inclusion of fermionic fields with spins higher than or equal to 3/2. This will require a classification of contraction procedures of Lie superalgebras and can lead to higher spin generalizations of three-dimensional Extended Bargmann supergravity [74].
Some of the results presented in this paper are also useful for studies of Hořava-Lifshitz gravity, that has been proposed as a new framework for Lifshitz holography [12,13,[90][91][92][93][94][95]. Extended Bargmann gravity has been argued to correspond to a special case of Hořava-Lifshitz gravity [13]. In this paper, we have constructed spin-3 generalizations of Extended Bargmann gravity. It is conceivable that these can be interpreted as suitable spin-3 generalizations of Hořava-Lifshitz gravity. It would be interesting to check whether this is indeed the case and whether the construction presented here can be generalized to yield spin-3 generalizations of generic Hořava-Lifshitz gravity theories.
Finally, higher spin theory has recently been argued to describe some of the excitations in fractional quantum Hall liquids [96]. Newton-Cartan geometry and gravity, that are based on extensions of the Galilei algebra, have been very useful in constructing effective actions that can capture transport properties in studies of the fractional quantum Hall effect. It would be interesting to investigate whether the non-relativistic higher spin gravity theories that can be constructed using the results of this paper, can play a similar role.

A Conventions
In this paper, we adopt the convention that the symmetrization of a pair of indices a, b are denoted with parentheses (ab), while anti-symmetrization is denoted with square brackets [ab]. Symmetrization and anti-symmetrization is performed without normalization factor, i.e., Nested (anti-)symmetrizations are understood to be taken from the outermost ones to the innermost ones, e.g.
Vertical bars denote that the (anti-)symmetrization does not affect the enclosed indices, e.g., For the Levi-Civita symbol, we adopt the following convention: The Lie algebraic direct sum of the Lie algebras a and b is denoted by a ⊕ b, i.e. a and b are ideals. The direct sum as vectorspaces is denoted by a + b, i.e. they do not necessarily commute with each other.

B Proof of theorem 1
In order to obtain contraction procedures of hs 3 AdS 3 and hs 3 dS 3 that reduce to those of table 1 when restricted to the spin-2 part, we start from the subalgebras h of table 1. For each of these four contraction procedures, one needs to add spin-3 generators to the subalgebra h, since otherwise one is led to contractions with an abelian spin-3 part. From table 1 one sees that the subalgebra h always contains the generator J. The spin-3 generators that one adds to h therefore need to fall into irreducible representations of J, under the adjoint action, in order to make sure that the enlarged h is a subalgebra. The spin-3 generators fall into the following irreducible representations of J:

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The proof then proceeds by checking, for each of the subalgebras h of table 1, which of these irreducible representations can be added to h, such that the enlarged h forms a subalgebra that leads to a contraction with a non-abelian spin-3 part. Below we discuss the different contraction procedures.
• Space-time contraction procedures: in this case we add irreducible representations (B.1) to {J, G a }.
-Adding J a , one finds from the commutator [G a , J b ] that one needs to add all G ab (i.e. both irreducible representations {G 12 , G 22 − G 11 } and {G 11 + G 22 }) in order to obtain a subalgebra. One thus finds the subalgebra h = {J, G a , J a , G ab }.
-Adding instead {G 12 , G 22 − G 11 }, one finds from the commutator of these two generators with G a that one also needs to add J a to obtain a subalgebra. From the commutator [G a , J b ], one then finds that one also needs to add all G ab . One thus again finds the subalgebra h = {J, G a , J a , G ab }.
-Similarly, if one adds {G 11 + G 22 } to {J, G a }, one is again led to the subalgebra h = {J, G a , J a , G ab }. We thus find that the only non-trivial space-time contraction procedures are the ones based on the subalgebras {J, G a , J a , G ab } and {J, G a , H a , P ab }.
• Speed-time contraction procedures: in this case we add irreducible representations (B.1) to {J, P a }. We can apply the same reasoning as for the space-time contraction procedures to find that the only non-trivial speed-time contraction procedures are the ones based on the subalgebras {J, P a , J a , P ab } and {J, P a , H a , G ab }.
• Speed-space contraction procedures: in this case we add irreducible representations (B.1) to {J, H}.
-Adding J a , one finds from the commutator [H, J a ] that one also needs to add H a to obtain a subalgebra. Similarly, adding instead H a , one finds from the commutator [H, H a ] that one needs to add J a to obtain a subalgebra. Adding either J a or H a thus leads to a subalgebra {J, H, J a , H a }. This subalgebra is maximal in the sense that adding any other spin-3 generators leads to a trivial contraction procedure.

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-Adding instead only G 11 + G 22 or P 11 + P 22 leads to a contraction with abelian spin-3 part, so this is excluded.
-Adding {G 12 , G 22 − G 11 }, one finds from their commutators with H that one also needs to add {P 12 , P 22 −P 11 }. Similarly, adding {P 12 , P 22 −P 11 }, one finds that one also needs to add {G 12 , G 22 − G 11 }. One thus finds a contraction procedure based on the subalgebra {J, H, G 12 , G 22 − G 11 , P 12 , P 22 − P 11 }. This subalgebra is not maximal as both G 11 + G 22  -Adding J a leads to a subalgebra {J, J a } that obeys all requirements.
-Adding H a similarly leads to a subalgebra {J, H a } that satisfies all requirements.
-Since [J a , H b ] generates H, adding J a and H a simultaneously brings one back to the case of the speed-space contraction procedures that was already discussed above.
-Adding extra spin-3 generators to either {J, J a } or {J, H a } leads only to trivial contractions.
-Adding only G 11 + G 22 or P 11 + P 22 to {J} leads to a contraction with abelian spin-3 part, so this is excluded.

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.