Drinfeld center and representation theory for monoidal categories

TL;DR

This paper explores the relationship between the Drinfeld center, representation theory, and property (T) in monoidal categories, providing new constructions and proofs, especially for categories of Hilbert bimodules over II$_1$-factors.

Contribution

It introduces a novel approach to constructing *-representations of fusion algebras in monoidal categories and proves property (T) invariance under weak monoidal Morita equivalence.

Findings

Constructed *-representations of fusion algebras from unitary half-braidings.

Proved the Drinfeld center of bimodule categories is monoidally equivalent to bimodule categories over a different II$_1$-factor.

Established the invariance of property (T) under weak monoidal Morita equivalence.

Abstract

Motivated by the relation between the Drinfeld double and central property (T) for quantum groups, given a rigid C*-tensor category C and a unitary half-braiding on an ind-object, we construct a *-representation of the fusion algebra of C. This allows us to present an alternative approach to recent results of Popa and Vaes, who defined C*-algebras of monoidal categories and introduced property (T) for them. As an example we analyze categories C of Hilbert bimodules over a II$_1$-factor. We show that in this case the Drinfeld center is monoidally equivalent to a category of Hilbert bimodules over another II$_1$-factor obtained by the Longo-Rehren construction. As an application, we obtain an alternative proof of the result of Popa and Vaes stating that property (T) for the category defined by an extremal finite index subfactor $N \subset M$ is equivalent to Popa's property (T) for the corresponding SE-inclusion of II$_1$-factors. In the last part of the paper we study M\"uger's notion of weakly monoidally Morita equivalent categories and analyze the behavior of our constructions under the equivalence of the corresponding Drinfeld centers established by Schauenburg. In particular, we prove that property (T) is invariant under weak monoidal Morita equivalence.