Centers, cocenters and simple quantum groups

TL;DR

This paper introduces a notion of centers for linearly reductive quantum groups and proves that quotients by these centers are simple under certain conditions, expanding understanding of quantum group structures.

Contribution

It defines the concept of a center for quantum groups and demonstrates simplicity of quotients by centers for various classes of quantum groups.

Findings

Quotients of free unitary groups by their centers are simple.

Quotients of quantum reflection groups by their centers are simple.

Several non-commutative quantum groups are shown to be simple.

Abstract

We define the notion of a (linearly reductive) center for a linearly reductive quantum group, and show that the quotient of a such a quantum group by its center is simple whenever its fusion semiring is free in the sense of Banica and Vergnioux. We also prove that the same is true of free products of quantum groups under very mild non-degeneracy conditions. Several natural families of compact quantum groups, some with non-commutative fusion semirings and hence very "far from classical", are thus seen to be simple. Examples include quotients of free unitary groups by their centers, recovering previous work, as well as quotients of quantum reflection groups by their centers.