Hopf algebra actions on strongly separable extensions of depth two

Details Hopf algebra actions on strongly separable extensions of depth two Hopf algebra actions on strongly separable extensions of depth two

2001-07-09

arxiv.org/abs/math/0107064v1

Adv. Math. 163 (2001), no. 2, 258--286

Mathematics

Rings and Algebras

21 pages, ams-latex; to appear in Advances in Mathematics

Scientific paper

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.

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