Abstract
ABSTRACT Suppose k is a field and is a separable Frobenius extension of k-algebras with trivial centralizer , Markov trace, and N a direct summand in M as N-bimodules. Let and be the successive endomorphism rings in a Jones tower (cf. Sec. 2). We define in Sec. 3 a depth 2 condition on this tower by requiring that a basis of freely generates as an M-module and a basis of freely generates as an -module. Then we provae in Sec. 4 that A and B have involutive strongly separable Hopf algebra structures dual to one another. As our main results, we prove in Sec. 5 that is a B-module algebra such that is the smash product ; in Sec. 6, that M is a A-module algebra such that is . We show that the actions involved are both outer. In Sec. 7, we prove that is a Hopf-Galois extension and point out a converse, thereby finding a non-commutative analogue of the classical theorem: a finite degree field extension is Galois if and only if it is separable and normal.
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