I-factorial quantum torsors and Heisenberg algebras of quantized universal enveloping type
arXiv:1702.08191 · doi:10.1016/j.jfa.2017.09.005
Abstract
We introduce a notion of I-factorial quantum torsor, which consists of an integrable ergodic action of a locally compact quantum group on a type I-factor such that also the crossed product is a type I-factor. We show that any such I-factorial quantum torsor is at the same time a I-factorial quantum torsor for the dual locally compact quantum group, in such a way that the construction is involutive. As a motivating example, we show that quantized compact semisimple Lie groups, when amplified via a crossed product construction with the function algebra on the associated weight lattice, admit I-factorial quantum torsors, and give an explicit realization of the dual quantum torsor in terms of a deformed Heisenberg algebra for the Borel part of a quantized universal enveloping algebra.
75 pages; the first four sections have appeared in the preprint arXiv:1612.00640, which will however not be published as such