2001/07/09 by Lars Kadison, Dmitri Nikshych
Mathematics · #math.RA #math.QA #msc:12F10 #msc:16W30 #msc:22D30 #msc:46L37
published as Adv. Math. 163 (2001), no. 2, 258--286 · 21 pages, ams-latex; to appear in Advances in Mathematics
arxiv created 2001/07/09 · arxiv updated 2009/11/30
We bring together ideas in analysis of Hopf *-algebra actions on II1 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 CM(N). Let M1 := End(M)N and M2 := End(M1)M be the endomorphism algebras in the Jones tower N < M < M1 < M2. We show that under depth 2 conditions on the second centralizers A := CM1(N) and B : = CM2(M) the algebras A and B are semisimple Hopf algebras dual to one another and such that M1 is a smash product of M and A, and that M is a B-Galois extension of N.