Conformal Newton–Hooke symmetry of Pais–Uhlenbeck oscillator

/pdf/transposed-poisson-structures-on-galilean-and-solvable-lie-1s52d0lg.pdf

Transposed Poisson structures on Galilean and solvable Lie algebras

We study the relationships among the various forms of the $q$ oscillator algebra and consider the conditions under which it supports a Hopf structure. We also present a generalization of this algebra together with its corresponding Hopf structure. Its multimode extensions are also considered.

Generalized q-Oscillators and their Hopf Structures

The analysis previously developed in [J. Math. Phys. 55 102901 (2014)] is used to construct systems which hold invariant under N=2l-conformal Galilei superalgebra. The models describe two different supersymmetric extensions of a free higher-derivative particle. Their Newton-Hooke counterparts are derived by applying appropriate coordinate transformations.

Higher-derivative mechanics with N=2l-conformal Galilei supersymmetry

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 that the Verma modules are reducible for vanishing highest weights. For half-integer \ell it is shown that all the Verma module is reducible. These results are independent of the central charges.

Galilean conformal algebras in two spatial dimension

l-Conformal Galilei algebra, denoted by g{l}{d}, 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 g{l}{d} with central extension as kinematical symmetry. This is done by developing a representation theory such as Verma modules, singular vectors of g{l}{d} and vector field representations for d = 1, 2.

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.

https://iopscience.iop.org/article/10.1088/1751-8113/46/40/405204/pdf

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

Representations of the quantum superalgebra U_q[osp(1/2)] and their relations to the basic hypergeometric functions are investigated. We first establish Clebsch-Gordan decomposition for the superalgebra U_q[osp(1/2)] in which the representations having no classical counterparts are incorporated. Formulae for these Clebsch-Gordan coefficients are derived, and it is observed that they may be expressed in terms of the $Q$-Hahn polynomials. We next investigate representations of the quantum supergroup OSp_q(1/2) which are not well-defined in the classical limit. Employing the universal T-matrix, the representation matrices are obtained explicitly, and found to be related to the little Q-Jacobi polynomials. Characteristically, the relation Q = -q is satisfied in all cases. Using the Clebsch-Gordan coefficients derived here, we construct new noncommutative spaces that are covariant under the coaction of the even dimensional representations of the quantum supergroup OSp_q(1/2).

Basic Hypergeometric Functions and Covariant Spaces for Even Dimensional Representations of U_q[osp(1/2)]

The conformal Galilei algebra (CGA) is a non-semisimple Lie algebra labelled by two parameters d and '. The aim of the present work is to investigate the lowest weight representations of CGA with d = 1 for any integer value of '. First we focus on the reducibi- lity of the Verma modules. We give a formula for the Shapovalov determinant and it follows that the Verma module is irreducible if ' = 1 and the lowest weight is nonvanishing. We prove that the Verma modules contain many singular vectors, i.e., they are reducible when '6 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.

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

We obtain a closed form expression of the universal T-matrix encapsulating the duality of the quantum superalgebra U_q[osp(1/2)] and the corresponding supergroup OSp_q(1/2). The classical q-->1 limit of this universal T-matrix yields the group element of the undeformed OSp(1/2) supergroup. The finite dimensional representations of the quantum supergroup OSp_q(1/2) are readily constructed employing the said universal T-matrix and the known finite dimensional representations of the dually related deformed U_q[osp(1/2)] superalgebra. Proceeding further, we derive the product law, the recurrence relations and the orthogonality of the representations of the quantum supergroup OSp_q(1/2). It is shown that the entries of these representation matrices are expressed in terms of the little Q-Jacobi polynomials with Q = -q. Two mutually complementary singular maps of the universal T-matrix on the universal R-matrix are also presented.

Universal T-matrix, Representations of OSp_q(1/2) and Little Q-Jacobi Polynomials

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