Connections between finite dimensional corepresentations of \(U_q(2)\) and \(S_qU(2)\) (Q1974166)

Let \(\mathbb{C}\) denote the field of complex numbers and let \(q\) be a non-zero real number. Let \(M_q(2)\) be the space of \(2\times 2\) quantum matrices over \(\mathbb{C}\) and let \(D\in M_q(2)\) denote the quantum determinant. In this paper the finite dimensional corepresentations of the Hopf \(*\)-algebras \(U_q(2)=M_q(2)[D^{-1}]\) and \(S_qU(2)\) are related, where \(S_qU(2)\) is the Hopf \(*\)-algebra constructed by \textit{S.~L.~Woronowicz} [in Publ. Res. Inst. Math. Sci. 23, No. 1, 117-181 (1987; Zbl 0676.46050)]. A new Hopf \(*\)-algebra \(U_q(2)[D^{1/2}]\) is constructed containing \(U_q(2)\) as a Hopf \(*\)-subalgebra, and endowed with an inclusion of Hopf \(*\)-algebras \(S_qU(2)\hookrightarrow U_q(2)[D^{1/2}]\). By this technique, it is shown that any finite dimensional corepresentation of \(U_q(2)\) can be decomposed into those of \(S_qU(2)\) and vice versa. Similar results for finite dimensional coalgebra morphisms are also obtained.