Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

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.