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

Abstract

We show that, for $q\in (0,1)$, the $C^{\ast }$-algebra ${\mathrm{SU}}_q(3)$ is isomorphic to a rank $2$ graph $C^{\ast }$-algebra (in the sense of Kumjian and Pask), and describe the graph in terms of its skeleton and commutation relations. Moreover, this isomorphism is ${\mathbb{T}}^2$-equivariant with respect to the right action on ${\mathrm{SU}}_q(3)$ and the gauge action coming from the $2$-graph.