Representations of the Quantum Algebra U q (u n , 1)

Abstract

The main aim of the paper is to study infinite-dimensional representations of the real form U q (u n, 1) of the quantized universal enveloping algebra U q (gl n + 1). We investigate the principal series of representations of U q (u n, 1) and calculate the intertwining operators for pairs of these representations. Some of the principal series representations are reducible. The structure of these representations is determined. Then we classify irreducible representations of U q (u n, 1) obtained from irreducible and reducible principal series representations. 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.