$C(S{O}_q(4)/S{O}_q(2))$ as a Groupoid $C^{\ast }$-algebra

Abstract

In this paper, we prove that $C(S{O}_q(4)/S{O}_q(2))$ is isomorphic to the $C^{\ast }$-algebra of the tight groupoid ${\mathcal{G}}_{\mathrm{tight}}$ associated with the inverse semigroup generated by the standard generators of its classical limit $C(S{O}_0(4)/S{O}_0(2))$. We show that all four orbits of the unit space ${\mathcal{G}}_{\mathrm{tight}}^{(0)}$ under the natural action of ${\mathcal{G}}_{\mathrm{tight}}$ are locally closed, and that the associated isotropy groups are isomorphic to $\mathbb{Z}$. Consequently, every irreducible representation of $C^{\ast }({\mathcal{G}}_{\mathrm{tight}})$ is induced from an irreducible representation of $C^{\ast }(\mathbb{Z})$, which are parametrized by $\mathbb{T}$. In this way, we obtain four families of irreducible representations parametrized by $\mathbb{T}$, and we explicitly construct their equivalence with the corresponding Soibelman irreducible representations of $C(S{O}_q(4)/S{O}_q(2))$.