Representations of the quantum algebra \(U_q({\mathfrak u}_{n,1})\) (Q1579886)

The complex quantum universal enveloping algebra \(U_q(\text{gl}_{n+1})\) has a real form \(U_q(\mu_{n,1})\). This paper studies infinite-dimensional representations of \(U_q(\mu_{n,1})\). The principal series of representations of \(U_q(\mu_{n,1})\) are studied, and the intertwining operators for pairs of such representations are calculated. Some of the principal series representations are reducible, and these are determined. Irreducible representations are required to have their restrictions to a maximal compact subalgebra \(U_q(u_n\oplus u_1)\) decompose as a direct sum of finite-dimensional irreducible representations of this subalgebra. The authors classify such irreducible representations of \(U_q(\mu_{n,1})\) obtained from irreducible and reducible principal series representations. All \(*\)-representations in this set of irreducible representations are separated. Unlike the classical case, the algebra \(U_q(\mu_{n,1})\) has finite-dimensional irreducible \(*\)-representations.