# I-factorial quantum torsors and Heisenberg algebras of quantized universal enveloping type

## TL;DR

This paper introduces I-factorial quantum torsors, exploring their properties for locally compact quantum groups, and provides explicit examples involving quantized Lie groups and deformed Heisenberg algebras.

## Contribution

It defines I-factorial quantum torsors, proves their duality properties, and constructs explicit examples using quantized Lie groups and deformed Heisenberg algebras.

## Key findings

- I-factorial quantum torsors are dual for the quantum group and its dual.
- Quantized compact semisimple Lie groups admit I-factorial quantum torsors.
- Explicit realization of dual torsors via deformed Heisenberg algebras.

## 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.