Kac−Paljutkin quantum group as a quantum subgroup of the quantum SU(2)

We show that the Kac−Paljutkin Hopf algebra appears as a quotient of \(C(\mathit{SU}_{-1}(2))\), which means that the corresponding quantum group \(G_{\kp}\) can be regarded as a quantum subgroup of \(\mathit{SU}_{-1}(2)\). 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.