FINITE-DIMENSIONAL SINGLETONS OF THE QUANTUM ANTI-DESITTER ALGEBRA

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 (N2 - 1)/2 and the Rac has dimension (N2 + 1)/2, while if N is even both the Di and Rac have dimension N2/2. 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).