Solvability of a Class of Braided Fusion Categories

SONIA NATALE; JULIA YAEL PLAVNIK

Pucón

Congreso; XIX Coloquio Latinoamericano de Algebra (CLA); 2012

In this talk we shall consider the general problem of understanding the structure of a fusion category C from the knowledge of the set c: d:(C) of Frobenius-Perron dimensions of its simple objects. We shall emphasize on the discussion of the notions of solvable and group-theoretical fusion categories, in the sense introduced by Etingof, Nikshych and Ostrik [1]. Some results of this type were obtained in the paper [3], where we have considered, for example, the case where c: d:(C) = f1; pg, with p an odd prime number. We shall show that a braided fusion category C such that its Frobenius-Perron dimen- sion is a natural number and every simple object of C has Frobenius-Perron dimension  2 is solvable. In addition, we shall also prove that such a fusion category is group- theoretical in the extreme case where the universal grading group of C is trivial. It is known that this conclusion is true in the opposite extreme case, that is, when C is a nilpotent braided fusion category that is also integral, which means that c: d:(C)  Z+. Some of our proofs rely in part on the results of Naidu and Rowell [2] for the case where C is integral and has a faithful self-dual simple object of Frobenius-Perron dimension 2. References [1] P. Etingof, D. Nikshych and V. Ostrik, Weakly group-theoretical and solvable fusion categories, Adv. Math. 226, 176205 (2011). [2] D. Naidu and E. Rowell, A niteness property for braided fusion categories, Algebr. Represent. Theory 14, 837855 (2011). [3] S. Natale and J. Plavnik, On fusion categories with few irreducibles degrees, Algebra and Number Theory 6, No. 6 (2012), 1171-1197.