Séminaire Algèbre Géométrie Algébrique Topologie Algébrique :

Le 19 septembre 2019 à 11:30 - salle 430

Présentée par Gainutdinov Azat - Universität Hamburg

On deformation of tensor categories

Tensor categories, on the one hand, are natural generalizations of Hopf algebras, and on the other hand, they give us a very convenient language both in representation theory and in mathematical physics. For example, many algebraic aspects of two-dimensional conformal field theories can be formulated in the language of tensor categories. I am interested in the problem of deformation of such categories and will talk about new results in this direction. The Hochschild type complexes called "Davydov-Yetter" classify infinitesimal deformations of tensor categories and of tensor functors. Our first result is that Davydov-Yetter cohomology for finite tensor categories is equivalent to the comonad cohomology of the central Hopf monad. This has several applications: First, we obtain a short and conceptual proof of Ocneanu rigidity. Second, it allows to use standard methods from comonad cohomology theory to compute Davydov-Yetter cohomology for a family of non-semisimple finite-dimensional Hopf algebras generalizing Sweedler's four dimensional Hopf algebra.