Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure

TL;DR

This paper explores a modified version of quantized enveloping algebras for compact simple Lie algebras, introducing a real structure that affects Verma modules and quantum homogeneous spaces in operator algebras.

Contribution

It introduces a new real-valued character modification to the quantized enveloping algebra and studies its impact on representation theory and quantum homogeneous spaces.

Findings

Development of a modified quantized enveloping algebra with real structure

Analysis of Verma modules and their quotients in the modified setting

Construction of examples of quantum homogeneous spaces in operator algebras

Abstract

Let $\mathfrak{g}$ be a compact simple Lie algebra. We modify the quantized enveloping $^*$-algebra associated to $\mathfrak{g}$ by a real-valued character on the positive part of the root lattice. We study the ensuing Verma module theory, and the associated quotients of these modified quantized enveloping $^*$-algebras. Restricting to the locally finite part by means of a natural adjoint action, we obtain in particular examples of quantum homogeneous spaces in the operator algebraic setting.