|M.Sc Student||Musicantov Evgeny|
|Subject||Extensions and Equivariantizations of Fusion Categories|
|Department||Department of Mathematics||Supervisor||Professor Shlomo Gelaki|
|Full Thesis text|
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])