Positive-energy irreps of the quantum anti de sitter algebra

We obtain positive-energy irreducible representations of the q-deformed anti de Si tter algebra U_q(so(3, 2)) by deformation of the classical ones. When the deformation parameter q is N-th root of unity, all these irreducible representations become unitary and finite-dimensional. Generically, their dimensions are smaller than those of the corresponding finite-dimensional non-unitary representations of so(3, 2). We discuss in detail the single-ton representations, i.e. the Di and Rac. When N is odd, the Di has dimension 1/2(N² - 1) and the Rac has dimension 1/2(N² + 1), while if N is even, both the Di and Rac have dimension 1/2N². 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).