Wreath products of finite groups by quantum groups

TL;DR

This paper introduces a new construction called partition wreath product combining finite groups and partition quantum groups, analyzes its properties, and computes its representation theory, especially for abelian groups.

Contribution

It defines the partition wreath product, relates it to classical and free wreath products, and characterizes its structure and representation theory.

Findings

Partition wreath product generalizes classical and free wreath products.

When the finite group is abelian, the product is a partition quantum group.

Representation theory of the constructed quantum group is explicitly computed.

Abstract

We introduce a notion of partition wreath product of a finite group by a partition quantum group, a construction motivated on the one hand by classical wreath products and on the other hand by the free wreath product of J. Bichon. We identify the resulting quantum group in several cases, establish some of its properties and show that when the finite group in question is abelian, the partition wreath product is itself a partition quantum group. This allows us to compute its representation theory, using earlier results of the first named author.