M.Sc Thesis
M.Sc Student Musicantov Evgeny Extensions and Equivariantizations of Fusion Categories Department of Mathematics Professor Shlomo Gelaki

Abstract

In this work we study fusion categories and their relations to semisimple Hopf algebras over an algebraically closed field k of characteristics 0. This relation is established via the Krein-Tannaka duality, which associates to a finite dimensional Hopf algebra H the category of its finite dimensional representations Rep(H) and the forgetful functor F to the category of finite dimensional k-vector spaces.

We concentrate our attention on p^n-dimensional semisimple Hopf algebras H, where p is an integer prime. In this case the fusion category Rep(H) is graded by the cyclic group on p elements C_p. It turns out that there is a p^(n-1)-dimensional semisimple Hopf algebra L such that D equivalent to Rep(L). This suggests an induction approach to the classification of p^n-dimensional semisimple Hopf algebras. That is, assuming the classification of p^(n-1)-dimensional semisimple Hopf algebras is already settled, one needs to classify C_p-extensions of Rep(L) for each p^(n-1)-dimensional semisimple Hopf algebra L and to classify fiber functors on these extensions. In [ENO] the authors classified all G-extensions of a fusion category D, where G is a finite group, in terms of homotopy types of maps from the classifying space of G to the classifying space of the Brauer-Picard groupoid of D.

The classification in [ENO] is done for general fusion categories. By restricting ourselves to the case of fusion categories with a fiber functor, we are able to prove that Rep(H) can be built as a C_p-equivariantization of Rep(L), where L is a p^(n-1)-dimensional semisimple Hopf algebra, in such a way that the forgetful functor on Rep(H) is induced by the forgetful functor of Rep(L).

Our main result is as follows. Let G be a finite group and let D be a fusion category. Suppose that C is a G-extension of D and N is an indecomposable C-module such that its restriction to D remains indecomposable. We prove that the dual category of C with respect to N is tensor equivalent to a G-equivariantization of the dual category of C with respect to N.

We apply the above result to the classification problem of p^n-dimensional semisimple Hopf algebras and completely classify p^3-dimensional Hopf algebras (recovering the classification result in [MA])