Finite-dimensional singletons of the quantum anti de Sitter algebra

Abstract

We obtain positive-energy irreducible representations of the q-deformed anti de Sitter algebra U q (so(3,2)) by deformation of the classical ones. When the deformation parameter q is an Nth root of unity, all these irreducible representations become unitary and finite-dimensional. Generically, their dimensions are smaller than of the corresponding finite-dimensional non-unitary representation of so(3,2). We discuss in detail the singleton representations, i.e. the Di and Rac. When N is odd the Di has dimension (N 2minus;1) 2 and the Rac has dimension (N 2+1) 2 , while if N is even both the Di and Rac have dimension N 2 2 . These dimensions are classical only for N=3 when the Di and Rac are deformations of the two fundamental non-unitary representations of so(3,2).