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; Theorem 2.7 corrected September 26, 2020

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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