Properties Of Slightly Degenerate Fusion Categories And Their Classifications

In this thesis,we first consider properties and structures of slightly degener-ate fusion categories and non-degenerate fusion categories of certain Frobenius-Perron dimensions.And then we classify slightly degenerate fusion categories of various Frobenius-Perron dimensions.In particular,we prove the minimal exten-sion conjecture for slightly degenerate weakly group-theoretical fusion categories.We also study the structure of non-degenerate weakly group-theoretical fusion category C,whose Frobenius-Perron dimension contains a square-free integer di-visor d.Under certain condition,we prove that C contains a non-degenerate fusion subcategory C(Zd,η)of Frobenius-Perron dimension d.The thesis contains three chapters.In the first chapter,we recall some basic notions and notations of fusion categories,such as adjoint fusion subcat-egories,Frobenius-Perron dimensions of fusion categories,graded fusion cate-gories,braided fusion categories,module categories over fusion categories,sym-metric fusion categories and Tannakian fusion categories,slightly degenerate and non-degenerate fusion categories,Morita equivalence of fusion category,weakly group-theoretical fusion categories and solvable fusion categories,etc.In the second chapter,we mainly classify weakly integral slightly degenerate fusion categories.First,for a weakly integral braided fusion category C,we show that if 8(?)FPdim(C)and Miiger center C’ is super-Tannakian then C is integral.Then we prove for a super-modular fusion category C that(?)is an algebraic integer for all X ∈O(C).Using these previous results,we classify slightly degenerate fusion categories of Frobenius-Perron dimensions 2pnd,4pnd,8d,16d and 2m with m ≤5,we also prove weakly group-theoretical property of braided fusion categories of Frobenius-Perron dimensions pmrnd,where p is an odd prime,r is a prime,d is a square-free integer such that(pr,d)=1.Particularly,we obtain that integral slightly degenerate fusion categories of Frobenius-Perron dimensions less than 64 are pointed.We also consider the structure of slightly degenerate generalized Tambara-Yamagami fusion categories.In the third chapter,we further consider non-degenerate and slightly degen-erate weakly group-theoretical fusion categories of Frobenius-Perron dimensions nd,where d is a square-free integer such that(n,d)=1.At first,given a non-degenerate weakly group-theoretical fusion category C with FPdim(C)=nd,if for all simple objects X ∈ C,we have(FPdim(X)2,d)=1,then we prove that C always contains a non-degenerate fusion category C(Zd,η),where C(Zd,叼)is a pointed non-degenerate fusion category corresponding to metric group with a non-degenerate quadratic form η.Therefore,C≌C(Zd,η)(?)C(Zd,η)’,C(Zd,η)’is the centralizer of C(Zd,η)in C.Then we also show that the minimal exten-sion conjecture of slightly degenerate weakly group-theoretical fusion categories is true.As a corollary,so we can generalize above conclusions about non-degenerate fusion categories to slightly degenerate cases.In particular,we obtain that inte-gral braided fusion category C of Frobenius-Perron dimension pnd with C’(?)sVec is nilpotent and group-theoretical,where p is a prime,d is a square-free integer such that(p,d)=1.In the last,we classify non-degenerate fusion categories of Frobenius-Perron dimensions p2q2d,p2q3d and p3q3d,where p,q are odd primes and is a square-free integer such that(pq,d)=1.