Unitary Representations and BRST Structure of the Quantum Anti--de Sitter Group at Roots of Unity

Abstract

It is shown that for suitable roots of unity, there exist finite--dimensional unitary representations of $U_q(so(2,3))$ corresponding to all classical one-particle representations with (half)integer spin, with the correct low-energy limit. In the massless case for spin $\geq 1$, a subspace of "pure gauges'' appears which must be factored out, as classically. Unitary many-particle representations are defined, with the same low-energy states as classically. Furthermore, a remarkable element of the center of $U_q(so(2,3))$ is identified which plays the role of the BRST operator, for any spin. The corresponding ghosts are an intrinsic part of indecomposable representations.