Modified trace is a symmetrised integral

TL;DR

This paper introduces a modified trace concept in pivotal categories, linking it to Calabi-Yau structures and providing a method to compute traces in quantum groups at roots of unity.

Contribution

It establishes a connection between non-degenerate modified traces and Calabi-Yau structures, and offers a way to compute these traces for quantum groups using integrals.

Findings

Modified trace defines a Calabi-Yau structure on projective objects.

Modified trace is determined by a symmetric linear form from an integral.

Computed modified traces for simply laced restricted quantum groups at roots of unity.

Abstract

A modified trace for a finite k-linear pivotal category is a family of linear forms on endomorphism spaces of projective objects which has cyclicity and so-called partial trace properties. We show that a non-degenerate modified trace defines a compatible with duality Calabi-Yau structure on the subcategory of projective objects. The modified trace provides a meaningful generalisation of the categorical trace to non-semisimple categories and allows to construct interesting topological invariants. We prove, that for any finite-dimensional unimodular pivotal Hopf algebra over a field k, a modified trace is determined by a symmetric linear form on the Hopf algebra constructed from an integral. More precisely, we prove that shifting with the pivotal element defines an isomorphism between the space of right integrals, which is known to be 1-dimensional, and the space of modified traces. This result allows us to compute modified traces for all simply laced restricted quantum groups at roots of unity.