### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 081, 20 pages      arXiv:1307.3642      https://doi.org/10.3842/SIGMA.2013.081
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

### Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure

Kenny De Commer
Department of Mathematics, University of Cergy-Pontoise, UMR CNRS 8088, F-95000 Cergy-Pontoise, France

Received August 18, 2013, in final form December 18, 2013; Published online December 24, 2013

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.

Key words: compact quantum homogeneous spaces; quantized universal enveloping algebras; Hopf-Galois theory; Verma modules.

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