Discrete series of unitary irreducible representations of the U q (u(3, 1)) and U q (u(n, 1)) noncompact quantum algebras

Abstract

The structure of all discrete series of unitary irreducible representations of the U q (u(3, 1)) and U q (u(n, 1)) noncompact quantum algebras are investigated with the aid of extremal projection operators and the q-analog of the Mickelsson-Zhelobenko algebra Z(g, g′) q . The orthonormal basis constructed in the infinite-dimensional space of irreducible representations of the U q (u(n, 1)) ⊇ U q (u(n)) algebra is the q-analog of the Gelfand-Graev basis in the space of the corresponding irreducible representations of the u(n, 1) ⊇ u(n) classical algebra.