Hopf Algebra Actions on Strongly Separable Extensions of Depth Two
✍ Scribed by Lars Kadison; Dmitri Nikshych
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- English
- Weight
- 245 KB
- Volume
- 163
- Category
- Article
- ISSN
- 0001-8708
No coin nor oath required. For personal study only.
✦ Synopsis
We bring together ideas in analysis on Hopf f-algebra actions on II 1 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 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 + M + M 1 + 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.