Normal subgroups and relative centers of linearly reductive quantum groups

TL;DR

This paper investigates the structure and representation theory of linearly reductive quantum groups, establishing conditions for normal subgroups, a Clifford-style correspondence, and a description of the relative center, generalizing classical results.

Contribution

It provides new structural and representation-theoretic results for linearly reductive quantum groups, including a center reconstruction theorem and conditions for normality.

Findings

Normal quantum subgroups are linearly reductive under certain antipode invariance conditions.

A Clifford-style correspondence exists between irreducible representations of quantum groups and their normal subgroups.

The relative center of quantum groups can be described by generators and relations, generalizing classical results.

Abstract

We prove a number of structural and representation-theoretic results on linearly reductive quantum groups, i.e. objects dual to that of cosemisimple Hopf algebras: (a) a closed normal quantum subgroup is automatically linearly reductive if its squared antipode leaves invariant each simple subcoalgebra of the underlying Hopf algebra; (b) for a normal embedding $\mathbb{H}\trianglelefteq \mathbb{G}$ there is a Clifford-style correspondence between two equivalence relations on irreducible $\mathbb{G}$- and, respectively, $\mathbb{H}$-representations; and (c) given an embedding $\mathbb{H}\le \mathbb{G}$ of linearly reductive quantum groups the Pontryagin dual of the relative center $Z(\mathbb{G})\cap \mathbb{H}$ can be described by generators and relations, with one generator $g_V$ for each irreducible $\mathbb{G}$-representation $V$ and one relation $g_U=g_Vg_W$ whenever $U$ and $V\otimes W$ are not disjoint over $\mathbb{H}$. This latter center-reconstruction result generalizes and recovers M\"uger's compact-group analogue and the author's quantum-group version of that earlier result by setting $\mathbb{H}=\mathbb{G}$.