Representations of the Quantum Algebra Uq(un, 1)

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.