A classification of $SU(d)$-type C$^*$-tensor categories

Kazhdan and Wenzl classified all rigid tensor categories with fusion ring isomorphic to the fusion ring of the group $SU(d)$. In this paper we consider the C$^*$-analogue of this problem. Given a rigid C$^*$-tensor category $\mathcal{C}$ with fusion ring isomorphic to the fusion ring of the group $SU(d)$, we can extract a constant $q$ from $\mathcal{C}$ such that there exists a $*$-representation of the Hecke algebra $H_n(q)$ into $\mathcal{C}$. The categorical trace on $\mathcal{C}$ induces a Markov trace on $H_n(q)$. Using this Markov trace and a representation of $H_n(q)$ in $\textrm{Rep}\,(SU_{\sqrt{q}}(d))$ we show that $\mathcal{C}$ is equivalent to a twist of the category $\textrm{Rep}\,(SU_{\sqrt{q}}(d))$. Furthermore a sufficient condition on a C$^*$-tensor category $\mathcal{C}$ is given for existence of an embedding of a twist of $\textrm{Rep}\,(SU_{\sqrt{q}}(d))$ in $\mathcal{C}$.