Galilei Algebra Research Articles

ℓ-oscillators from second-order invariant PDEs of the centrally extended conformal Galilei algebras

We construct, for any given ℓ=12+N0, the second-order, linear partial differential equations (PDEs) which are invariant under the centrally extended conformal Galilei algebra. At the given ℓ, two invariant equations in one time and ℓ+12 space coordinates are obtained. The first equation possesses a continuum spectrum and generalizes the free Schrödinger equation (recovered for ℓ=12) in 1 + 1 dimension. The second equation (the “ℓ-oscillator”) possesses a discrete, positive spectrum. It generalizes the 1 + 1-dimensional harmonic oscillator (recovered for ℓ=12). The spectrum of the ℓ-oscillator, derived from a specific osp(1|2ℓ + 1) h.w.r., is explicitly presented. The two sets of invariant PDEs are determined by imposing (representation-dependent) on-shell invariant conditions both for degree 1 operators (those with continuum spectrum) and for degree 0 operators (those with discrete spectrum). The on-shell condition is better understood by enlarging the conformal Galilei algebras with the addition of certain second-order differential operators. Two compatible structures (the algebra/superalgebra duality) are defined for the enlarged set of operators.

Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras

The conformal Galilei algebra (CGA) is a non-semisimple Lie algebra labelled by two parameters $d$ and $\ell$. The aim of the present work is to investigate the lowest weight representations of CGA with $d = 1$ for any integer value of $\ell$. First we focus on the reducibility of the Verma modules. We give a formula for the Shapovalov determinant and it follows that the Verma module is irreducible if $\ell = 1$ and the lowest weight is nonvanishing. We prove that the Verma modules contain many singular vectors, i.e., they are reducible when $\ell \neq 1$. Using the singular vectors, hierarchies of partial differential equations defined on the group manifold are derived. The differential equations are invariant under the kinematical transformation generated by CGA. Finally we construct irreducible lowest weight modules obtained from the reducible Verma modules.

Dynamical realizations of non-relativistic conformal groups

Dynamical realizations of the l-conformal Galilei algebra and its Newton-Hooke counterpart in terms of second order differential equations are discussed.

Conformal Newton–Hooke algebras, Niederer's transformation and Pais–Uhlenbeck oscillator

Dynamical systems invariant under the action of the l-conformal Newton–Hooke algebras are constructed by the method of nonlinear realizations. The relevant first order Lagrangians together with the corresponding Hamiltonians are found. The relation to the Galajinsky and Masterov [24] approach as well as the higher derivatives formulation is discussed. The generalized Niederer's transformation is presented which relates the systems under consideration to those invariant under the action of the l-conformal Galilei algebra [25]. As a nice application of these results an analogue of Niederer's transformation, on the Hamiltonian level, for the Pais–Uhlenbeck oscillator is constructed.

Dynamical realizations of $\mathcal {N}=1$N=1 l-conformal Galilei superalgebra

Dynamical systems which are invariant under \documentclass[12pt]{minimal}\begin{document}$\mathcal {N}=1$\end{document}N=1 supersymmetric extension of the l-conformal Galilei algebra are constructed. These include a free \documentclass[12pt]{minimal}\begin{document}$\mathcal {N}=1$\end{document}N=1 superparticle which is governed by higher derivative equations of motion and an \documentclass[12pt]{minimal}\begin{document}$\mathcal {N}=1$\end{document}N=1 supersymmetric Pais-Uhlenbeck oscillator for a particular choice of its frequencies. A Niederer-like transformation which links the models is proposed.

Representations of ℓ-conformai Galilei algebra and hierarchy of invariant equation

The ℓ-conformai Galilei algebra, denoted by gℓ(d), is a particular non-semisimple Lie algebra specified by a positive integer d and a spin value ℓ. The algebra gℓ(d) admits central extensions. We study lowest weight representations, in particular Verma modules, of gℓ(d) with the central extensions for d = 1,2. We give a classification of irreducible modules over d = 1 algebras and a condition of the Verma modules over d = 2 algebras being reducible. As an application of the representation theory, hierarchies of differential equations are derived. The Lie group generated by gℓ(d) with the central extension is a kinematical symmetry of the differential equations.

On simple modules over conformal Galilei algebras

We study irreducible representations of two classes of conformal Galilei algebras in 1-spatial dimension. We construct a functor which transforms simple modules with nonzero central charge over the Heisenberg subalgebra into simple modules over the conformal Galilei algebras. This can be viewed as an analogue of oscillator representations. We use oscillator representations to describe the structure of simple highest weight modules over conformal Galilei algebras. We classify simple weight modules with finite dimensional weight spaces over finite dimensional Heisenberg algebras and use this classification and properties of oscillator representations to classify simple weight modules with finite dimensional weight spaces over conformal Galilei algebras.

Chiral and Real N = 2 supersymmetric -conformal Galilei algebras

dx.doi.org/10.7437/NF 2318-4957 /2013.01.003 Inequivalent N = 2 supersymmetrizations of the -conformal Galilei algebra ind-spatial dimensions are constructed from the chiral (2, 2) and the real (1, 2, 1)basic supermultiplets of the N = 2 supersymmetry. For non-negative integer andhalf-integer both superalgebras admit a consistent truncation with a (different)finite number of generators. The real N = 2 case coincides with the superalgebraintroduced by Masterov, while the chiral N = 2 case is a new superalgebra.We present D-module representations of both superalgebras. Then we investi-gate the new superalgebra derived from the chiral supermultiplet. It is shown that itadmits two types of central extensions, one is found for any d and half-integer andthe other only for d = 2 and integer . For each central extension the centrally ex-tended -superconformal Galilei algebra is realized in terms of its super-Heisenbergsubalgebra generators.

Intertwining operators for ℓ-conformal Galilei algebras and hierarchy of invariant equations

The ℓ-conformal Galilei algebra, denoted by is a non-semisimple Lie algebra specified by a pair of parameters (d, ℓ). The algebra is regarded as a nonrelativistic analogue of the conformal algebra. We derive hierarchies of partial differential equations which have invariance of the group generated by with a central extension as kinematical symmetry. This is done by developing a representation theory such as Verma modules, singular vectors of and vector field representations for d = 1, 2.

Chiral and real ${\cal N}=2$N=2 supersymmetric ℓ-conformal Galilei algebras

Inequivalent \documentclass[12pt]{minimal}\begin{document}${\cal N}=2$\end{document}N=2 supersymmetrizations of the ℓ-conformal Galilei algebra in d-spatial dimensions are constructed from the chiral (2, 2) and the real (1, 2, 1) basic supermultiplets of the \documentclass[12pt]{minimal}\begin{document}${\cal N}=2$\end{document}N=2 supersymmetry. For non-negative integer and half-integer ℓ, both superalgebras admit a consistent truncation with a (different) finite number of generators. The real \documentclass[12pt]{minimal}\begin{document}${\cal N}=2$\end{document}N=2 case coincides with the superalgebra introduced by Masterov, while the chiral \documentclass[12pt]{minimal}\begin{document}${\cal N}=2$\end{document}N=2 case is a new superalgebra. We present D-module representations of both superalgebras. Then we investigate the new superalgebra derived from the chiral supermultiplet. It is shown that it admits two types of central extensions, one is found for any d and half-integer ℓ, and the other only for d = 2 and integer ℓ. For each central extension, the centrally extended ℓ-superconformal Galilei algebra is realized in terms of its super-Heisenberg subalgebra generators.

Leibniz homology of the Galilei algebra

In this paper, we compute the Leibniz homology of the Galilei algebra.

‘Stringy’ Newton–Cartan gravity

We construct a ‘stringy’ version of Newton–Cartan gravity in which the concept of a Galilean observer plays a central role. We present both the geodesic equations of motion for a fundamental string and the bulk equations of motion in terms of a gravitational potential which is a symmetric tensor with respect to the longitudinal directions of the string. The extension to include a nonzero cosmological constant is given. We stress the symmetries and (partial) gaugings underlying our construction. Our results provide a convenient starting point to investigate applications of the AdS/CFT correspondence based on the non-relativistic ‘stringy’ Galilei algebra.

HIGHEST WEIGHT REPRESENTATIONS AND KAC DETERMINANTS FOR A CLASS OF CONFORMAL GALILEI ALGEBRAS WITH CENTRAL EXTENSION

We investigate the representations of a class of conformal Galilei algebras in one spatial dimension with central extension. This is done by explicitly constructing all singular vectors within the Verma modules, proving their completeness and then deducing irreducibility of the associated highest weight quotient modules. A resulting classification of infinite dimensional irreducible modules is presented. It is also shown that a formula for the Kac determinant is deduced from our construction of singular vectors. Thus we prove a conjecture of Dobrev, Doebner and Mrugalla for the case of the Schrödinger algebra.

Dynamical realization of l-conformal Galilei algebra and oscillators

The method of nonlinear realizations is applied to the l-conformal Galilei algebra to construct a dynamical system without higher derivative terms in the equations of motion. A configuration space of the model involves coordinates, which parametrize particles in d spatial dimensions, and a conformal mode, which gives rise to an effective external field. It is shown that trajectories of the system can be mapped into those of a set of decoupled oscillators in d dimensions.

$\mathcal {N}=2$ N = 2 supersymmetric extension of l-conformal Galilei algebra

\documentclass[12pt]{minimal}\begin{document}$\mathcal {N}=2$\end{document} N = 2 supersymmetric extension of the l-conformal Galilei algebra is constructed. A relation between its representations in flat spacetime and in Newton-Hooke spacetime is discussed. An infinite-dimensional generalization of the superalgebra is given.

Generalized relativistic kinematics

We propose a method for deforming an extended Galilei algebra that leads to a nonstandard realization of the Poincar\'e group with the Fock-Lorentz linear fractional transformations. The invariant parameter in these transformations has the dimension of length. Combining this deformation with the standard one (with an invariant velocity $c$) leads to the algebra of the symmetry group of the anti-de Sitter space in Beltrami coordinates. In this case, the action for free point particles contains the dimensional constants $R$ and $c$. The limit transitions lead to the ordinary ($R\to \infty$) or alternative ($c\to \infty$) but nevertheless relativistic kinematics.

On irreducible representations of the exotic conformal Galilei algebra

We investigate the representations of the exotic conformal Galilei algebra. This is done by explicitly constructing all singular vectors within the Verma modules, and then deducing irreducibility of the associated highest weight quotient modules. A resulting classification of infinite-dimensional irreducible modules is presented.

Comments on Galilean conformal field theories and their geometric realization

We discuss non-relativistic conformal algebras generalizing the Schr\"odinger 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 Euclidean. We comment on holographic-type calculations in this background.

The exotic conformal Galilei algebra and nonlinear partial differential equations

The conformal Galilei algebra ( cga) and the exotic conformal Galilei algebra ( ecga) are applied to construct partial differential equations (PDEs) and systems of PDEs, which admit these algebras. We show that there are no single second-order PDEs invariant under the cga but systems of PDEs can admit this algebra. Moreover, a wide class of nonlinear PDEs exists, which are conditionally invariant under cga. It is further shown that there are systems of nonlinear PDEs admitting ecga with the realisation obtained very recently in [D. Martelli, Y. Tachikawa, Comments on Galilei conformal field theories and their geometric realisation, preprint, arXiv:0903.5184v2 [hep-th], 2009]. Moreover, wide classes of nonlinear systems, invariant under two different 10-dimensional subalgebras of ecga are explicitly constructed and an example with possible physical interpretation is presented.

Can cosmic acceleration be caused by exotic massless particles?

To describe dark energy we introduce a fluid model with no free parameter on the microscopic level. The constituents of this fluid are massless particles which are a dynamical realization of the unextended $D=(3+1)$ Galilei algebra. These particles are exotic as they live in an enlarged phase space. Their only interaction is with gravity. A minimal coupling to the gravitational field, satisfying Einstein's equivalence principle, leads to a dynamically active gravitational mass density of either sign. A two-component model containing matter (baryonic and dark) and dark energy leads, through the cosmological principle, to Friedmann-like equations. Their solutions show a deceleration phase for the early universe and an acceleration phase for the late universe. We predict the Hubble parameter $H(z)/{H}_{0}$ and the deceleration parameter $q(z)$ and compare them with available experimental data. We also discuss a reduced model (one-component dark sector) and the inclusion of radiation. Our model shows no stationary modification of Newton's gravitational potential.