Modified trace, integrals and invariants (Azat Gainutdinov, Université de Tours)

17.05.2018 14:15

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. It is also turned out to be useful in analysis of mapping class group representations. Our main theorem says that for any finite-dimensional unimodular pivotal Hopf algebra H over a field, a modified trace is determined by a symmetric linear form on H 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 allowed us to compute modified traces for all simply laced restricted quantum groups at roots of unity. This is a joint work with Anna Beliakova and Christian Blanchet.

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Room 17, Séminaire "Topologie et Géométrie"

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