Hopf Algebra Actions on Strongly Separable Extensions of Depth Two

Abstract

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↪M is a separable Frobenius extension of k-algebras with trivial centralizer CM(N) and split as N-bimodules. Let M1≔End(MN) and M2≔End(M1)M be the endomorphism algebras in the Jones tower N↪M↪M1 ↪M2. We place depth 2 conditions on its second centralizers A≔CM1(N) and B≔CM2(M). We prove that A and B are semisimple Hopf algebras dual to one another, that M1 is a smash product of M and A, and that M is a B-Galois extension of N.