Integrals for finite tensor categories

TL;DR

This paper extends the concepts of integrals and cointegrals from finite-dimensional Hopf algebras to finite tensor categories, establishing foundational results and exploring their properties and applications.

Contribution

It introduces categorical integrals and cointegrals, generalizes key properties from Hopf algebras, and extends the notion of indicators to finite tensor categories.

Findings

Existence and uniqueness of categorical integrals and cointegrals

Generalization of Maschke's theorem to tensor categories

Extension of the $n$-th indicator concept to tensor categories

Abstract

We introduce the notions of categorical integrals and categorical cointegrals of a finite tensor category $\mathcal{C}$ by using a certain adjunction between $\mathcal{C}$ and its Drinfeld center $\mathcal{Z}(\mathcal{C})$. These notions can be identified with integrals and cointegrals of a finite-dimensional Hopf algebra $H$ if $\mathcal{C}$ is the representation category of $H$. We generalize basic results on integrals and cointegrals of a finite-dimensional Hopf algebra (such as the existence, the uniqueness, and the Maschke theorem) to finite tensor categories. Motivated by results of Lorenz, we also investigate relations between categorical integrals and morphisms factoring through projective objects. Finally, we extend the $n$-th indicator of a finite-dimensional Hopf algebra introduced by Kashina, Montgomery and Ng to finite tensor categories.