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Journal Articles Selecta Math. Year : 2021

Modified trace is a symmetrised integral


A modified trace for a finite ${\mathbb {k}}$-linear pivotal category is a family of linear forms on endomorphism spaces of projective objects which has cyclicity and so-called partial trace properties. The modified trace provides a meaningful generalisation of the categorical trace to non-semisimple categories and allows to construct interesting topological invariants. We show that a non-degenerate modified trace defines a compatible with duality Calabi–Yau structure on the subcategory of projective objects. We prove, that for any finite-dimensional unimodular pivotal Hopf algebra over a field ${\mathbb {k}}$, 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.

Dates and versions

hal-03224696 , version 1 (11-05-2021)



Anna Beliakova, Christian Blanchet, Azat M. Gainutdinov. Modified trace is a symmetrised integral. Selecta Math., 2021, 27 (3), pp.31. ⟨10.1007/s00029-021-00626-5⟩. ⟨hal-03224696⟩
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