Non-extremal weight modules for quantized universal enveloping algebras

TL;DR

This paper constructs and analyzes non-extremal weight modules for quantized universal enveloping algebras, focusing on induced modules from the centralizer and their relation to unitary representations of non-compact real forms.

Contribution

It introduces a method to construct weight modules via induction from the centralizer, including finite-dimensional and one-dimensional modules, and explores their properties and connections to unitary representations.

Findings

Induced modules from the centralizer are well-behaved and structurally rich.

Special case of $U_q(sl(2,C))$ yields admissible unitary representations of $U_q(su(1,1)).

Relation established between induced modules and commutative subalgebras of the centralizer.

Abstract

For quantized universal enveloping algebras we construct weight modules by inducing representations of the centralizer of the Cartan subalgebra in the quantized universal enveloping algebra. The induced modules arising from finite-dimensional weight modules the centralizer algebra are studied. In particular, we study the induction of one-dimensional modules, and this is related to the study of commutative subalgebras of the centralizer algebra. For the special case of $U_q(\mathfrak{sl}(2,\mathbb{C}))$ we show that we get the admissible unitary representations corresponding to the non-compact real form $U_q(\mathfrak{su}(1,1))$.