Quantum $SU(3)$ as the $C^{\ast }$-algebra of a 2-graph

Abstract

We show that for $q\in (0,1),$ the $C^{\ast }$-algebra $S{U}_q(3)$ is isomorphic a rank $2$ graph $C^{\ast }$-algebra (in the sense of Pask and Kumjian). This graph is derived by passing the to the limit $q\to 0$ for a set of generators of $S{U}_q(3)$. Moreover, the isomorphism can be taken to be ${\mathbb{T}}^2$-equivariant with respect the right-action on $S{U}_q(3)$ and the gauge action coming from the $2$-graph