Quantum $SU(3)$ as the $C^*$-algebra of a 2-graph

TL;DR

This paper demonstrates that the quantum group $SU_q(3)$ for $q$ in (0,1) can be represented as a $C^*$-algebra of a 2-graph, linking quantum groups with graph algebra structures.

Contribution

It establishes an isomorphism between $SU_q(3)$ and a 2-graph $C^*$-algebra, derived via a limit process as $q$ approaches zero, and preserves certain symmetries.

Findings

$SU_q(3)$ is isomorphic to a 2-graph $C^*$-algebra for $q o 0$

The isomorphism is $ ext{T}^2$-equivariant

The construction links quantum groups with graph algebra frameworks

Abstract

We show that for $q\in (0,1),$ the $C^{*}$-algebra $SU_{q}(3)$ is isomorphic a rank $2$ graph $C^{*}$-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 $SU_{q}(3)$. Moreover, the isomorphism can be taken to be $\mathbb{T}^{2}$-equivariant with respect the right-action on $SU_{q}(3)$ and the gauge action coming from the $2$-graph