Infinite-dimensional Galilean conformal algebras can be constructed by contracting pairs of symmetry algebras in conformal field theory, such as $W$-algebras. Known examples include contractions of pairs of the Virasoro algebra, its $N=1$ superconformal extension, or the $W_3$ algebra. Here, we introduce a contraction prescription of the corresponding operator-product algebras, or equivalently, a prescription for contracting tensor products of vertex algebras. With this, we work out the Galilean conformal algebras arising from contractions of $N=2$ and $N=4$ superconformal algebras as well as of the $W$-algebras $W(2,4)$, $W(2,6)$, $W_4$, and $W_5$. The latter results provide evidence for the existence of a whole new class of $W$-algebras which we call Galilean $W$-algebras. We also apply the contraction prescription to affine Lie algebras and find that the ensuing Galilean affine algebras admit a Sugawara construction. The corresponding central charge is level-independent and given by twice the dimension of the underlying finite-dimensional Lie algebra. Finally, applications of our results to the characterisation of structure constants in $W$-algebras are proposed.
A Galilean contraction is a way to construct Galilean conformal algebras from a pair of infinite-dimensional conformal algebras, or equivalently, a method for contracting tensor products of vertex algebras. Here, we present a generalisation of the Galilean contraction prescription to allow for inputs of any finite number of conformal algebras, resulting in new classes of higher-order Galilean conformal algebras. We provide several detailed examples, including infinite hierarchies of higher-order Galilean Virasoro algebras, affine Kac-Moody algebras and the associated Sugawara constructions, an…
Galilean conformal algebras can be constructed by contracting a finite number of conformal algebras, and enjoy truncated $\mathbb{Z}$-graded structures. Here, we present a generalisation of the Galilean contraction procedure, giving rise to Galilean conformal algebras with truncated $\mathbb{Z}^{\otimesσ}$-gradings, $σ\in\mathbb{N}$. Detailed examples of these multi-graded Galilean algebras are provided, including extensions of the Galilean Virasoro and affine Kac-Moody algebras. We also derive the associated Sugawara constructions and discuss how these examples relate to multivariable extensi…
We consider the non-relativistic c -> \infty contraction limit of the (N=2k)- extended D=4 superconformal algebra su(2,2;N), introducing in this way the non-relativistic (N=2k)-extended Galilean superconformal algebra. Such a Galilean superconformal algebra has the same number of generators as su(2,2|2k). The usp(2k) algebra describes the non-relativistic internal symmetries, and the generators from the coset u(2k)/usp(2k) become central charges after contraction.
Firstly we discuss briefly three different algebras named as nonrelativistic (NR) conformal: Schroedinger, Galilean conformal and infinite algebra of local NR conformal isometries. Further we shall consider in some detail Galilean conformal algebra (GCA) obtained in the limit c equal to infinity from relativistic conformal algebra O(d+1,2) (d - number of space dimensions). Two different contraction limits providing GCA and some recently considered realizations will be briefly discussed. Finally by considering NR contraction of D=4 superconformal algebra the Galilei conformal superalgebra (GCSA…
We propose canonical and Lie-algebraic twist deformations of $κ$-deformed Poincare Hopf algebra which leads to the generalized $κ$-Minkowski space-time relations. The corresponding deformed $κ$-Poincare quantum groups are also calculated. Finally, we perform the nonrelativistic contraction limit to the corresponding twisted Galilean algebras and dual Galilean quantum groups.
The way to obtain massive non-relativistic states from the Poincare algebra is twofold. First, following Inonu and Wigner the Poincare algebra has to be contracted to the Galilean one. Second, the Galilean algebra is to be extended to include the central mass operator. We show that the central extension might be properly encoded in the non-relativistic contraction. In fact, any Inonu-Wigner contraction of one algebra to another, corresponds to an infinite tower of abelian extensions of the latter. The proposed method is straightforward and holds for both central and non-central extensions. Apa…
We note that large classes of contractions of algebras that arise in physics can be understood purely algebraically, via identifying appropriate $\mathbb{Z}_m$-gradings (and their generalizations) on the parent algebra. This includes various types of flat space/Carroll limits of finite and infinite dimensional (A)dS algebras, as well as Galilean and Galilean Conformal algebras. Our observations can be regarded as providing a natural context for the Grassmann approach of arXiv:1312.2941. We also introduce a related notion, which we call partial grading, that arises naturally in this context.
The usual Galilean contraction procedure for generating new conformal symmetry algebras takes as input a number of symmetry algebras which are equivalent up to central charge. We demonstrate that the equivalence condition can be relaxed by inhomogeneously contracting the chiral algebras and present general results for the ensuing asymmetric Galilean algebras. Several examples relevant to conformal field theory are discussed in detail, including superconformal algebras and W-algebras. We also discuss how the Sugawara construction is modified in the asymmetric setting.
It is shown that the N-Galilean conformal algebra, with N-odd, is the maximal symmetry algebra of the free Lagrangian involving (N+1)/2-th order time derivative.
It is shown that centrally extended N-Galilean conformal algebra, with N-odd, is the maximal symmetry algebra of the Schrodinger equation corresponding to the free Lagrangian involving (N+1)/2-th order time derivatives.
Representation theory of an infinite dimensional Galilean conformal algebra introduced by Martelli and Tachikawa is developed. We focus on the algebra defined in (2+1) dimensional spacetime and consider central extension. It is then shown that the Verma modules are irreducible for non-vanishing highest weights. This is done by explicit computation of Kac determinant. We also present coadjoint representations of the Galilean conformal algebra and its Lie group. As an application of them, a coadjoint orbit of the Galilean conformal group is given in a simple case.
The semisimple part of d-dimensional Galilean conformal algebra g^(d) is given by h^(d)=O(2,1)+O(d), which after adding via semidirect sum the 3d-dimensional Abelian algebra t^(d) of translations, Galilean boosts and constant accelerations completes the construction. We obtain Galilean superconformal algebra G^(d) by firstly defining the semisimple superalgebra H^(d) which supersymmetrizes h^(d), and further by considering the expansion of H^(d) by tensorial and spinorial graded Abelian charges in order to supersymmetrize the Abelian generators of t^(d). For d=3 the supersymmetrization of h^(3…
We discuss non-relativistic conformal algebras generalizing the Schrödinger algebra. One instance of these algebras is a conformal, acceleration-extended, Galilei algebra, which arises also as a contraction of the relativistic conformal algebra. In two dimensions, this admits an "exotic" central extension, whereby boosts do not commute. We study general properties of non-relativistic conformal field theories with such symmetry. We realize geometrically the symmetry in terms of a metric invariant under the exotic conformal Galilei algebra, although its signature is neither Lorentzian nor Euclid…
We generalize the differential representation of the operators of the Galilean algebras to include fractional derivatives. As a result a whole new class of scale invariant Galilean algebras are obtained. The first member of this class has dynamical index $z=2$ similar to the Schrödinger algebra. The second member of the class has dynamical index $z=3/2$, which happens to be the dynamical index Kardar-Parisi-Zhang equation.
Galilean conformal algebra (GCA) is an Inonu-Wigner (IW) contraction of a conformal algebra, while Newton-Hooke string algebra is an IW contraction of an AdS algebra which is the isometry of an AdS space. It is shown that the GCA is a boundary realization of the Newton-Hooke string algebra in the bulk AdS. The string lies along the direction transverse to the boundary, and the worldsheet is AdS_2. The one-dimensional conformal symmetry so(2,1) and rotational symmetry so(d) contained in the GCA are realized as the symmetry on the AdS_2 string worldsheet and rotational symmetry in the space tran…
We find an intriguing link between the symmetries of the tensionless limit of closed string theory and the 2-dimensional Galilean Conformal Algebra (2d GCA). 2d GCA has been discussed in the context of the non-relativistic limit of AdS/CFT and more recently in flat-space holography as the proposed symmetry algebra of the field theory dual to 3d Minkowski spacetimes. It is best understood as a contraction of two copies of the Virasoro algebra. In this note, we link this to the tensionless limit of bosonic closed string theory. We show how it emerges naturally as a contraction of the residual ga…
Logarithmic representations of the conformal Galilean algebra (CGA) and the Exotic Conformal Galilean algebra ({\sc ecga}) are constructed. This can be achieved by non-decomposable representations of the scaling dimensions or the rapidity indices, specific to conformal galilean algebras. Logarithmic representations of the non-exotic CGA lead to the expected constraints on scaling dimensions and rapidities and also on the logarithmic contributions in the co-variant two-point functions. On the other hand, the {\sc ecga} admits several distinct situations which are distinguished by different sets…
We define a theory of Galilean gravity in 2+1 dimensions with cosmological constant as a Chern-Simons gauge theory of the doubly-extended Newton-Hooke group, extending our previous study of classical and quantum gravity in 2+1 dimensions in the Galilean limit. We exhibit an r-matrix which is compatible with our Chern-Simons action (in a sense to be defined) and show that the associated bi-algebra structure of the Newton-Hooke Lie algebra is that of the classical double of the extended Heisenberg algebra. We deduce that, in the quantisation of the theory according to the combinatorial quantisat…
A class of infinite dimensional Galilean conformal algebra in (2+1) dimensional spacetime is studied. Each member of the class, denoted by \alg_{\ell}, is labelled by the parameter \ell. The parameter \ell takes a spin value, i.e., 1/2, 1, 3/2, .... We give a classification of all possible central extensions of \alg_{\ell}. Then we consider the highest weight Verma modules over \alg_{\ell} with the central extensions. For integer \ell we give an explicit formula of Kac determinant. It results immediately that the Verma modules are irreducible for nonvanishing highest weights. It is also shown …
N = 2 Supersymmetric extensions of Galilean conformal algebra (GCA), specified by spin \ell and dimension of space d, are investigated. Duval and Horvathy showed that the \ell = 1/2 GCA has two types of supersymmetric extensions, called standard and exotic. Recently, Masterov intorduced a centerless super-GCA for arbitrary \ell wchich corresponds to the standard extension. We show that the Masterov's super-GCA has two types of central extensions depending on the parity of 2\ell. We then introduced a novel super-GCA for arbitrary \ell corresponding to the exotic extension. It is shown that the …
We introduce three nested Lie algebras of infinitesimal `isometries' of a Galilei space-time structure which play the rôle of the algebra of Killing vector fields of a relativistic Lorentz space-time. Non trivial extensions of these Lie algebras arise naturally from the consideration of Newton-Cartan-Bargmann automorphisms.
We investigate the existence of action for both the electric and magnetic sectors of Galilean Electrodynamics using Helmholtz conditions. We prove the existence of unique action in magnetic limit with the addition of a scalar field in the system. The check also implies the non existence of action in the electric sector of Galilean electrodynamics. Dirac constraint analysis of the theory reveals that there are no local degrees of freedom in the system. Further, the theory enjoys a reduced but an infinite dimensional subalgebra of Galilean conformal symmetry algebra as global symmetries. The ful…
In the present work we considered Galilean conformal algebras (GCA), which arises as a contraction relativistic conformal algebras ($x_i\rightarrow εx_i$, $t\rightarrow t$, $ε\rightarrow 0$). We can use the Galilean conformal symmetry to constrain two-point and three-point functions. Correlation functions in space-time without boundary condition were found in \cite{1}. In real situations there are boundary conditions in space-time, so we have calculated correlation functions for Galilean confrormal invariant fields in semi-infinite space with boundary condition in $r=0$. We have calculated two…
We consider Schrodinger equations for a non-relativistic particle obeying N+1-th order higher derivative classical equation of motion. These equations are invariant under N(odd)-extended Galilean conformal (NGC) algebras in general d+1 dimensions. In 2+1 dimensions, the exotic Schrodinger equations are invariant under N(even)-GCA.
We give an 8-dimensional realization of the Clifford algebra in the 5-dimensional Galilean space-time by using a dimensional reduction from the $(5+1)$ Minkowski space-time to the $(4+1)$ Minkowski space-time which encompasses the Galilean space-time. A set of solutions of the Dirac-type equation in the 5-dimensional Galilean space-time is obtained, based on the Pauli representation of $8\times 8$ gamma matrices. In order to find an explicit solution, we diagonalize the Klein-Gordon divisor by using the Galilean boost.
The Galilean conformal algebra has recently been realised in the study of the non-relativistic limit of the AdS/CFT conjecture. This was obtained by a systematic parametric group contraction of the parent relativistic conformal field theory. In this paper, we extend the analysis to include supersymmetry. We work at the level of the co-ordinates in superspace to construct the N=1 Super Galilean conformal algebra. One of the interesting outcomes of the analysis is that one is able to naturally extend the finite algebra to an infinite one. This looks structurally similar to the N=1 superconformal…
We apply the nonlinear realizations method for constructing new Galilean conformal mechanics models. Our starting point is the Galilean conformal algebra which is a non-relativistic contraction of its relativistic counterpart. We calculate Maurer-Cartan one-forms, examine various choices of the relevant coset spaces and consider the geometric inverse Higgs-type constraints which reduce the number of the independent coset parameters and, in some cases, provide dynamical equations. New Galilean conformally invariant actions are derived in arbitrary space-time dimension D=d+1 (no central charges)…
The non-relativistic versions of the generalized Poincaré algebras and generalized $AdS$-Lorentz algebras are obtained. This non-relativistic algebras are called, generalized Galilean algebras type I and type II and denoted by $\mathcal{G}\mathfrak{B}_{n}$ and $\mathcal{G}\mathfrak{L}_{_{n}}$ respectively. Using a generalized Inönü--Wigner contraction procedure we find that the generalized Galilean algebras type I can be obtained from the generalized Galilean algebras type II. The $S$-expansion procedure allows us to find the $\mathcal{G}\mathfrak{B}_{_{5}}$ algebra from the Newton--Hooke alge…
In this note, we construct a Wess-Zumino-Witten model based on the Galilean conformal algebra in 2-spacetime dimensions, which is a nonrelativistic analogue of the relativistic conformal algebra. We obtain exact background corresponding to σ-models in six dimensions (the dimension of the group manifold) and a central charge c=6. We carry out a Sugawara type construction to verify the conformal invariance of the model. Further, we discuss the feasibility of the background obtained as a physical spacetime metric.
Using the notion of Galilean conformal algebra (GCA) in arbitrary space dimension d, we introduce for d=3 quantized nonrelativistic counterpart of twistors as the spinorial representation of O(2,1){\oplus}SO(3) which is the maximal semisimple subalgebra of three-dimensional GCA. The GC-covariant quantization of such nonrelativistic spinors, which shall be called also Galilean twistors, is presented. We consider for d=3 the general spinorial matrix realizations of GCA, which are further promoted to quantum-mechanical operator representations, expressed as bilinears in quantized Galilean twistor…
Galilean Conformal Algebras (GCA) have been recently proposed as a different non-relativistic limit of the AdS/CFT conjecture. In this note, we look at the representations of the GCA. We also construct explicitly the two and three point correlators in this non-relativistic limit of CFT and comment on the differences with the relativistic case and also the more studied Schrodinger group.
In this paper, we construct a single Lagrangian for both limits of Galilean electrodynamics. The framework relies on a covariant formalism used in describing Newton-Cartan geometry. We write down the Galilean conformal algebra and its representation in this formalism. We also show that the Lagrangian is invariant under the Galilean conformal algebra in d = 4 and calculate the energy-momentum tensor.
We construct free Lie algebras which, together with the algebra of spatial rotations, form infinite-dimensional extensions of finite-dimensional Galilei Maxwell algebras appearing as global spacetime symmetries of extended non-relativistic objects and non-relativistic gravity theories. We show how various extensions of the ordinary Galilei algebra can be obtained by truncations and contractions, in some cases via an affine Kac-Moody algebra. The infinite-dimensional Lie algebras could be useful in the construction of generalized Newton-Cartan theories gravity theories and the objects that coup…
We study the space-time symmetries of the actions obtained by expanding the action for a massive free relativistic particle around the Galilean action. We obtain all the point space-time symmetries of the post-Galilean actions by working in canonical space. We also construct an infinite collection of generalized Schrödinger algebras parameterized by an integer $M$, with $M=0$ corresponding to the standard Schrödinger algebra. We discuss the Schrödinger equations associated to these algebras, their solutions and projective phases.
The six-dimensional exotic Galilean algebra in (2+1) dimensions with two central charges $m$ and $θ$, is extended when $m=0$, to a ten-dimensional Galilean conformal algebra with dilatation, expansion, two acceleration generators and the central charge $θ$. A realisation of such a symmetry is provided by a model with higher derivatives recently discussed in \cite{peterwojtek}. We consider also a realisation of the Galilean conformal symmetry for the motion with a Coulomb potential and a magnetic vortex interaction. Finally, we study the restriction, as well as the modification, of the Galilean…
We show that a class of nonrelativistic algebras including non centrally-extended Schrodinger algebra and Galilean Conformal Algebra (GCA) has an affine extension in 2+1 hitherto unknown. This extension arises out of the conformal symmetries of the two dimensional complex plain. We suggest that this affine form may be the symmetry that explains the relaxation of some classical phenomena towards their critical point. This affine algebra admits a central extension and maybe realized in the bulk. The bulk realization suggests that this algebra may be derived by looking at the asymptotic symmetry …
We characterise Lie groups with bi-invariant bargmannian, galilean or carrollian structures. Localising at the identity, we show that Lie algebras with ad-invariant bargmannian, carrollian or galilean structures are actually determined by the same data: a metric Lie algebra with a skew-symmetric derivation. This is the same data defining a one-dimensional double extension of the metric Lie algebra and, indeed, bargmannian Lie algebras coincide with such double extensions, containing carrollian Lie algebras as an ideal and projecting to galilean Lie algebras. This sets up a canonical correspond…
The Galilean Conformal Algebra (GCA) arises from the relativistic conformal algebra in the non-relativistic limit. In two dimensions, one can view it as a limit of linear combinations of the two copies Virasoro algebra. Recently, it has been argued that Topologically Massive Gravity (TMG) realizes the quantum 2d GCA in a particular scaling limit of the gravitational Chern-Simons term. To add strength to this claim, we demonstrate a matching of correlation functions on both sides of this correspondence. A priori looking for spatially dependent correlators seems to force us to deal with high spi…
Galilean conformal algebra (GCA) in two dimensions arises as contraction of two copies of the centrally extended Virasoro algebra ($t\rightarrow t, x\rightarrowεx$ with $ε\rightarrow 0$). The central charges of GCA can be expressed in term of Virasoro central charges. For finite and non-zero GCA central charges, the Virasoro central charges must behave as asymmetric form $O(1)\pm O(\frac{1}ε)$. We propose that, the bulk description for 2d GCA with asymmetric central charges is given by general massive gravity (GMG) in three dimensions. It can be seen that, if the gravitational Chern-Simons cou…
We study supersymmetric extensions of classical kinematical algebras from the point of view of contraction theory. It is shown that contracting the supersymmetric extension of the anti-de Sitter algebra leads to a hierarchy similar in structure to the classical Bacry-Lévy-Leblond classification
In this paper, we discuss the Inönü-Winger contraction of the conformal algebra. We start with the light-cone form of the Poincaré algebra and extend it to write down the conformal algebra in $d$ dimensions. To contract the conformal algebra, we choose five dimensions for simplicity and compactify the third transverse direction in to a circle of radius $R$ following Kaluza-Klein dimensional reduction method. We identify the inverse radius, $1/R$, as the contraction parameter. After the contraction, the resulting representation is found to be the continuous spin representation in four dimension…
We study the Galilean symmetry in a nonrelativistic model, recently advanced by Bak, Jackiw and Pi, involving the coupling of a nonabelian Chern-Simons term with matter fields. The validity of the Galilean algebra on the constraint surface is demonstrated in the gauge independent formalism. Then the reduced space formulation is discussed in the axial gauge using the symplectic method. An anomalous term in the Galilean algebra is obtained which can be eliminated by demanding conditions on the Green function. Finally, the axial gauge is also treated by Dirac's method. Galilean symmetry is preser…
Conformal Galilei algebra contains so(1,2) subalgebra which is the conformal algebra in one dimension. In this note we generalize methods previously developed for one-dimensional many-body systems and construct a unitary map relating a quantum mechanics invariant under the conformal Galilean transformations to a set of decoupled particles for which the same symmetry is realized in a nonlocal way. Possible applications of the map are discussed.
The scattering equation formalism for scattering amplitudes, and its stringy incarnation, the ambitwistor string, remains a mysterious construction. In this paper, we pursue the study a gauged-unfixed version of the ambitwistor string known as the null string. We explore the following three aspects in detail; its complexification, gauge fixing, and amplitudes. We first study the complexification of the string; the associated symmetries and moduli, and connection to the ambitwistor string. We then look in more details at the leftover symmetry algebra of the string, called Galilean conformal alg…
We study a new contraction of a d+1 dimensional relativistic conformal algebra where n+1 directions remain unchanged. For n=0,1 the resultant algebras admit infinite dimensional extension containing one and two copies of Virasoro algebra, respectively. For n> 1 the obtained algebra is finite dimensional containing an so(2,n+1) subalgebra. The gravity dual is provided by taking a Newton-Cartan like limit of the Einstein's equations of AdS space which singles out an AdS_{n+2} spacetime. The infinite dimensional extension of n=0,1 cases may be understood from the fact that the dual gravities cont…
Non-relativistic versions of the AdS/CFT conjecture have recently been investigated in some detail. These have primarily been in the context of the Schrodinger symmetry group. Here we initiate a study based on a {\it different} non-relativistic conformal symmetry: one obtained by a parametric contraction of the relativistic conformal group. The resulting Galilean conformal symmetry has the same number of generators as the relativistic symmetry group and thus is different from the Schrodinger group (which has fewer). One of the interesting features of the Galilean Conformal Algebra is that it a…
In the present paper we consider the realization of $2-$dimensional Galilean conformal algebra ($GCA_2$) on the boundary of cosmological new massive gravity. At first we consider the contracted BTZ black hole solution. We obtain entropy formula for the $GCA_2$ in term of contracted scaling dimension $Δ$ and central charge $C_1$. This entropy formula exactly matches with the non-relativistic limit of Bekenstein-Hawking entropy of BTZ black hole. Then we extend our study to the contracted warped $AdS_3$ black hole solution of CNMG. We obtain the entropy of dual $GCA_2$ in terms of central charge…
We show that an extended $3D$ Schrödinger algebra introduced in [1] can be reformulated as a $3D$ Poincaré algebra extended with an SO(2) R-symmetry generator and an $SO(2)$ doublet of bosonic spin-1/2 generators whose commutator closes on $3D$ translations and a central element. As such, a non-relativistic Chern-Simons theory based on the extended Schrödinger algebra studied in [1] can be reinterpreted as a relativistic Chern-Simons theory. The latter can be obtained by a contraction of the $SU(1,2)\times SU(1,2)$ Chern-Simons theory with a non principal embedding of $SL(2,\mathbb R)$ into $S…
Lie bialgebra contractions are introduced and classified. A non-degenerate coboundary bialgebra structure is implemented into all pseudo-orthogonal algebras $so(p,q)$ starting from the one corresponding to $so(N+1)$. It allows to introduce a set of Lie bialgebra contractions which leads to Lie bialgebras of quasi-orthogonal algebras. This construction is explicitly given for the cases $N=2,3,4$. All Lie bialgebra contractions studied in this paper define Hopf algebra contractions for the Drinfel'd-Jimbo deformations $U_z so(p,q)$. They are explicitly used to generate new non-semisimple quantum…
We present higher-spin algebras containing a Poincaré subalgebra and with the same set of generators as the Lie algebras that are relevant to Vasiliev's equations in any space-time dimension $D \geq 3$. Given these properties, they can be considered either as candidate rigid symmetries for higher-spin gauge theories in Minkowski space or as Carrollian conformal higher-spin symmetries in one less dimension. We build these Lie algebras as quotients of the universal enveloping algebra of $iso(1,D-1)$ and we show how to recover them as Inönü-Wigner contractions of the rigid symmetries of higher-sp…
We show the twisted Galilean invariance of the noncommutative parameter, even in presence of space-time noncommutativity. We then obtain the deformed algebra of the Schrödinger field in configuration and momentum space by studying the action of the twisted Galilean group on the non-relativistic limit of the Klein-Gordon field. Using this deformed algebra we compute the two particle correlation function to study the possible extent to which the previously proposed violation of the Pauli principle may impact at low energies. It is concluded that any possible effect is probably well beyond detect…
We consider a realization of the Galilean conformal algebra (GCA) in two dimensional space-time on the AdS boundary of a particular three dimensional gravity theory, the so-called cosmological topologically massive gravity (CTMG), which includes the gravitational Chern-Simons term and the negative cosmological constant. The infinite dimensional GCA in two dimensions is obtained from the Virasoro algebra for the relativistic CFT by taking a scaling limit $t\to t$, $x\toεx$ with $ε\to 0$. The parent relativistic CFT should have left and right central charges of order $\mathcal{O}(1/ε)$ but oppos…
We add to Galilean symmetries the transformations describing constant accelerations. The corresponding extended Galilean algebra allows, in any dimension $D=d+1$, the introduction of one central charge $c$ while in $D=2+1$ we can have three such charges: c, θand θ'. We present nonrelativistic classical mechanics models, with higher order time derivatives and show that they give dynamical realizations of our algebras. The presence of central charge $c$ requires the acceleration square Lagrangian term. We show that the general Lagrangian with three central charges can be reinterpreted as describ…
