# Kac-Paljutkin Quantum Group as a Quantum Subgroup of the Quantum SU(2)

## TL;DR

This paper demonstrates that the Kac-Paljutkin quantum group can be realized as a quantum subgroup of the deformed quantum group $SU_{-1}(2)$ by constructing an explicit Hopf *-homomorphism using categorical and algebraic methods.

## Contribution

It establishes the embedding of the Kac-Paljutkin quantum group into $SU_{-1}(2)$ via a quotient construction and graded twisting techniques.

## Key findings

- Kac-Paljutkin algebra is a quotient of $C(SU_{-1}(2))$
- Corepresentation category is a Tambara-Yamagami tensor category
- Constructed explicit Hopf *-homomorphism

## Abstract

We show that the Kac-Paljutkin Hopf algebra appears as a quotient of $C(SU_{-1}(2))$, which means that the corresponding quantum group $G_{KP}$ can be regarded as a quantum subgroup of $SU_{-1}(2)$. We combine the fact that corepresentation category of the Kac-Paljutkin Hopf algebra is a Tambara-Yamagami tensor category associated with the Krein 4-group and the method of graded twisting of Hopf algebras, to construct the Hopf *-homomorphism.