A categorical approach to dynamical quantum groups

TL;DR

This paper develops a categorical framework for dynamical quantum groups using Harish-Chandra bimodules, establishing duality theorems and connecting classical and quantum groups through functors and symmetries.

Contribution

It introduces a new categorical perspective on dynamical quantum groups, including Tannaka duality and the relation to parabolic restriction functors.

Findings

Proves Tannaka duality theorems for dynamical quantum groups.

Connects standard dynamical quantum groups to parabolic restriction functors.

Identifies a Weyl symmetry via Zhelobenko operators leading to the dynamical Weyl group action.

Abstract

We present a categorical point of view on dynamical quantum groups in terms of categories of Harish-Chandra bimodules. We prove Tannaka duality theorems for forgetful functors into the monoidal category of Harish-Chandra bimodules in terms of a slight modification of the notion of a bialgebroid. Moreover, we show that the standard dynamical quantum groups $F(G)$ and $F_q(G)$ are related to parabolic restriction functors for classical and quantum Harish-Chandra bimodules. Finally, we exhibit a natural Weyl symmetry of the parabolic restriction functor using Zhelobenko operators and show that it gives rise to the action of the dynamical Weyl group.