Modified trace is a symmetrised integral

Understand

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.