Representations of quantum algebra U_q(u_{n,1})

Abstract

Infinite dimensional representations of the real form U_q(u_{n,1}) of the Drinfeld--Jimbo algebra U_q(gl_{n+1}) are defined. The principal series of representations of U_q(u_{n,1}) is studied. Intertwining operators for pairs of the principal series representations are calculated in an explicit form. The structure of reducible representations of the principal series is determined. Irreducible representations of U_q(u_{n,1}), obtained from irreducible and reducible principal series representations, are classified. All *-representations in this set of irreducible representations are separated. Unlike the classical case, the algebra U_q(u_{n,1}) has finite dimensional irreducible *-representations.