Hopf algebra actions on strongly separable extensions of depth two

We bring together ideas in analysis on Hopf *-algebra actions on II1 subfactors of finite Jones index [9, 24] and algebraic characterizations of Frobenius, Galois and cleft Hopf extensions [3, 13, 14] to prove a non-commutative algebraic analogue of the classical theorem: a finite degree field extension is Galois iff it is separable and normal. Suppose N hooked right arrow M is a separable Frobenius extension of k-algebras with trivial centralizer C-M(N) and split as N-bimodules. Let M-1 : = End(M-N) and M-2 : = End(M-1)(M) be the endomorphism algebras in the Jones tower N hooked right arrow M hooked right arrow M-1 hooked right arrow M-2. We place depth 2 conditions on its second centralizers A : = C-M1 (N) and B: = C-M2 (M). We prove that A and B are semisimple Hopf algebras dual to one another. that M-1 is a smash product of M and A. and that M is a B-Galois extension of N. (C) 2001 Academic Press.