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

Abstract

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 𝒞 with fusion ring isomorphic to the fusion ring of the group SU(d), we can extract a constant q from 𝒞 such that there exists a *-representation of the Hecke algebra Hn(q) into 𝒞. The categorical trace on 𝒞 induces a Markov trace on Hn(q). Using this Markov trace and a representation of Hn(q) in [Formula: see text] we show that 𝒞 is equivalent to a twist of the category [Formula: see text]. Furthermore a sufficient condition on a C*-tensor category 𝒞 is given for existence of an embedding of a twist of [Formula: see text] in 𝒞.