# Categorical aspects of cointegrals on quasi-Hopf algebras

## TL;DR

This paper explores the relationship between category-theoretical notions and cointegrals in quasi-Hopf algebras, providing explicit descriptions and connections to Frobenius structures and module traces.

## Contribution

It offers explicit descriptions of categorical cointegrals, relates Frobenius structures to cointegrals in the unimodular case, and characterizes twisted module traces via cointegrals.

## Key findings

- Explicit description of categorical cointegrals in terms of algebra cointegrals
- Expression of Frobenius structures using cointegrals and integrals
- Description of twisted module trace in terms of cointegrals

## Abstract

We discuss relations between some category-theoretical notions for a finite tensor category and cointegrals on a quasi-Hopf algebra. Specifically, for a finite-dimensional quasi-Hopf algebra $H$, we give an explicit description of categorical cointegrals of the category ${}_H \mathscr{M}$ of left $H$-modules in terms of cointegrals on $H$. Provided that $H$ is unimodular, we also express the Frobenius structure of the `adjoint algebra' in the Yetter-Drinfeld category ${}^H_H \mathscr{YD}$ by using an integral in $H$ and a cointegral on $H$. Finally, we give a description of the twisted module trace for projective $H$-modules in terms of cointegrals on $H$.