The monoidal center and the character algebra

TL;DR

This paper develops a categorical character theory for pivotal finite tensor categories, defining class functions and internal characters via the monoidal center, extending Hopf algebra results, and exploring semisimplicity and character tables.

Contribution

It introduces a new algebra of class functions and internal characters for pivotal finite tensor categories, extending classical Hopf algebra character theory to a categorical setting.

Findings

The internal character map is an injective algebra homomorphism.

The map is an isomorphism if and only if the category is semisimple.

The character table relates to the S-matrix in modular tensor categories.

Abstract

For a pivotal finite tensor category $\mathcal{C}$ over an algebraically closed field $k$, we define the algebra $\mathsf{CF}(\mathcal{C})$ of class functions and the internal character $\mathsf{ch}(X) \in \mathsf{CF}(\mathcal{C})$ for an object $X \in \mathcal{C}$ by using an adjunction between $\mathcal{C}$ and its monoidal center $\mathcal{Z}(\mathcal{C})$. We also develop the integral theory in a unimodular finite tensor category by using the same adjunction. By utilizing these tools, we extend some results in the character theory of finite-dimensional Hopf algebras to this category-theoretical setting. Our main result is that the map $\mathsf{ch}: \mathsf{Gr}_k(\mathcal{C}) \to \mathsf{CF}(\mathcal{C})$ given by taking the internal character is a well-defined injective algebra map, where $\mathsf{Gr}_k(\mathcal{C})$ is the scalar extension of the Grothendieck ring of $\mathcal{C}$ to $k$. Moreover, under the assumption that $\mathcal{C}$ is unimodular, the map $\mathsf{ch}$ is an isomorphism if and only if $\mathcal{C}$ is semisimple. As an application, we show that the algebra $\mathsf{Gr}_{k}(\mathcal{C})$ is semisimple if $\mathcal{C}$ is a non-degenerate pivotal fusion category. If, moreover, $\mathsf{Gr}_k(\mathcal{C})$ is commutative, then the character table of $\mathcal{C}$ is defined based on the integral theory. It turns out that the character table is obtained from the $S$-matrix if $\mathcal{C}$ is a modular tensor category. Generalizing corresponding results in the finite group theory, we prove the orthogonality relations and the integrality.