Traces on Module Categories over Fusion Categories

TL;DR

This paper studies traces on module categories over pivotal fusion categories, showing they uniquely characterize Morita classes of certain Frobenius algebras and induce pivotal structures on dual categories.

Contribution

It introduces the concept of module traces compatible with the module structure and demonstrates their role in classifying Morita equivalence classes of Frobenius algebras.

Findings

Module traces are unique up to scale.

They characterize Morita classes of special haploid symmetric Frobenius algebras.

They induce pivotal structures on dual categories.

Abstract

We consider traces on module categories over pivotal fusion categories which are compatible with the module structure. It is shown that such module traces characterise the Morita classes of special haploid symmetric Frobenius algebras. Moreover, they are unique up to a scale factor and they equip the dual category with a pivotal structure. This implies that for each pivotal structure on a fusion category over the complex numbers there exists a conjugate pivotal structure defined by the canonical module trace.