Classifying braidings on fusion categories

We show that braidings on a fusion category $\mathcal{C}$ correspond to certain fusion subcategories of the center of $\mathcal{C}$ transversal to the canonical Lagrangian algebra. This allows to classify braidings on non-degenerate and group-theoretical fusion categories.Comment: 14 pages, minor corrections, new references adde

Hopf algebra actions on strongly separable extensions of depth two

We bring together ideas in analysis of Hopf *-algebra actions on II_1 subfactors of finite Jones index and algebraic characterizations of Frobenius, Galois and cleft Hopf extensions to prove a non-commutative algebraic analogue of the classical theorem: a finite field extension is Galois iff it is separable and normal. Suppose N < M is a separable Frobenius extension of k-algebras split as N-bimodules with a trivial centralizer C_M(N). Let M_1 := End(M)_N and M_2 := End(M_1)_M be the endomorphism algebras in the Jones tower N < M < M_1 < M_2. We show that under depth 2 conditions on the second centralizers A := C_{M_1}(N) and B : = C_{M_2}(M) the algebras A and B are semisimple Hopf algebras dual to one another and such that M_1 is a smash product of M and A, and that M is a B-Galois extension of N.Comment: 21 pages, ams-latex; to appear in Advances in Mathematic