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.Comment: 11 pages, added 6 references and conclusing remark

On realizations of polynomial algebras with three generators via deformed oscillator algebras

We present the most general polynomial Lie algebra generated by a second order integral of motion and one of order M, construct the Casimir operator, and show how the Jacobi identity provides the existence of a realization in terms of deformed oscillator algebra. We also present the classical analog of this construction for the most general Polynomial Poisson algebra. Two specific classes of such polynomial algebras are discussed that include the symmetry algebras observed for various 2D superintegrable systems.Comment: 28 page