The $C^*$-algebra of the quantum symplectic sphere

TL;DR

This paper demonstrates that the $C^*$-algebra of the quantum symplectic sphere is isomorphic to a graph $C^*$-algebra, linking it to quantum spheres and extending previous representation results.

Contribution

It shows the $C^*$-algebra of the quantum symplectic sphere is isomorphic to a graph $C^*$-algebra, generalizing prior work and connecting to quantum spheres.

Findings

$C^*(S_q^{4n-1})$ is isomorphic to a graph $C^*$-algebra

$C^*(S_q^{4n-1})$ is isomorphic to the quantum $(2(n+1)-1)$-sphere

First $n-1$ generators are zero in the algebra

Abstract

The faithful irreducible $*$-representations of the $C^*$-algebra of the quantum symplectic sphere $S_q^{4n-1}, n\geq 2$, have been investigated by D'Andrea and Landi. They proved that the first $n-1$ generators are all zero inside $C^*(S_q^{4n-1})$, for $n\geq 2$. The result is a generalisation of the case where $n=2$, which was shown by Mikkelsen and Szyma\'nski. We will show that $C^*(S_q^{4n-1}), n\geq 2$ is isomorphic to a graph $C^*$-algebra. From here it follows that $C^*(S_q^{4n-1})$ is isomorphic to the quantum $(2(n+1)-1)$-sphere by Vaksman and Soibelman.