On unimodular finite tensor categories

TL;DR

This paper characterizes unimodular finite tensor categories through several equivalent conditions involving adjoint functors and duality preservation, providing insights into their structure and applications to topological invariants.

Contribution

It establishes multiple equivalent conditions for unimodularity in finite tensor categories, linking categorical properties with duality and Frobenius functors, and applies these results to topological invariants.

Findings

Unimodularity is equivalent to the forgetful functor being Frobenius.

Conditions for duality preservation by adjoint functors are equivalent to unimodularity.

Application to topological invariants from unimodular Hopf algebras.

Abstract

Let $\mathcal{C}$ be a finite tensor category with simple unit object, let $\mathcal{Z}(\mathcal{C})$ denote its monoidal center, and let $L$ and $R$ be a left adjoint and a right adjoint of the forgetful functor $U: \mathcal{Z}(\mathcal{C}) \to \mathcal{C}$. We show that the following conditions are equivalent: (1) $\mathcal{C}$ is unimodular, (2) $U$ is a Frobenius functor, (3) $L$ preserves the duality, (4) $R$ preserves the duality, (5) $L(1)$ is self-dual, and (6) $R(1)$ is self-dual, where $1 \in \mathcal{C}$ is the unit object. We also give some other equivalent conditions. As an application, we give a categorical understanding of some topological invariants arising from finite-dimensional unimodular Hopf algebras.