Quantum automorphism groups of finite quantum groups are classical

TL;DR

This paper proves that the universal quantum automorphism group of a finite quantum group is actually a classical group, using multiplicative unitary operators to establish the result.

Contribution

It demonstrates that the quantum automorphism group of a finite quantum group is always classical, resolving a question about the nature of these automorphism groups.

Findings

Universal quantum automorphism groups are classical groups.

Use of multiplicative unitary operators in the proof.

Clarification of automorphism group structures for finite quantum groups.

Abstract

In a recent paper of Bhowmick, Skalski and So{\l}tan the notion of a quantum group of automorphisms of a finite quantum group was introduced and, for a given finite quantum group G, existence of the universal quantum group acting on G by automorphisms was proved. We show that this universal quantum group is in fact a classical group. The key ingredient of the proof is the use of multiplicative unitary operators, and we include a thorough discussion of this notion in the context of finite quantum groups.