Galilean conformal algebras in two spatial dimension

Abstract

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.