# Modified traces for quasi-Hopf algebras

## TL;DR

This paper establishes a correspondence between modified traces and cointegrals in finite-dimensional unimodular pivotal quasi-Hopf algebras, extending known results from Hopf to quasi-Hopf algebras and providing explicit computations for symplectic fermion examples.

## Contribution

It generalizes the theory of modified traces and cointegrals from Hopf algebras to the broader class of quasi-Hopf algebras, including explicit calculations.

## Key findings

- Non-zero modified traces are non-degenerate.
- Modified traces exist only for unimodular quasi-Hopf algebras.
- Explicit cointegrals and traces computed for symplectic fermion quasi-Hopf algebras.

## Abstract

Let H be a finite-dimensional unimodular pivotal quasi-Hopf algebra over a field k, and let H-mod be the pivotal tensor category of finite-dimensional H-modules. We give a bijection between left (resp. right) modified traces on the tensor ideal H-pmod of projective modules and left (resp. right) cointegrals for H. The non-zero left/right modified traces are non-degenerate, and we show that non-degenerate left/right modified traces can only exist for unimodular H. This generalises results of Beliakova, Blanchet, and Gainutdinov from Hopf algebras to quasi-Hopf algebras. As an example we compute cointegrals and modified traces for the family of symplectic fermion quasi-Hopf algebras.