$C(SO_q(4)/SO_q(2))$ as a Groupoid $C^*$-algebra

TL;DR

This paper demonstrates that the quantum homogeneous space $C(SO_q(4)/SO_q(2))$ can be realized as a groupoid $C^*$-algebra, providing a detailed analysis of its irreducible representations and their parametrization.

Contribution

It establishes an isomorphism between $C(SO_q(4)/SO_q(2))$ and a groupoid $C^*$-algebra, and explicitly constructs its irreducible representations.

Findings

All four orbits of the unit space are locally closed.

Isotropy groups are isomorphic to $bZ$.

Irreducible representations are parametrized by $bT$.

Abstract

In this paper, we prove that $C(SO_q(4)/SO_q(2))$ is isomorphic to the $C^*$-algebra of the tight groupoid $\mathcal{G}_{\mathrm{tight}}$ associated with the inverse semigroup generated by the standard generators of its classical limit $C(SO_0(4)/SO_0(2))$. 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 $C^*(\mathcal{G}_{\mathrm{tight}})$ is induced from an irreducible representation of $C^*(\mathbb{Z})$, 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 $C(SO_q(4)/SO_q(2))$.