OUTER ACTIONS OF CENTRALIZER HOPF ALGEBRAS ON SEPARABLE EXTENSIONS

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.

ACKNOWLEDGMENT

The authors thank J. Brzezinski, J. Cuntz, E. Effros, S. Montgomery, P. Schauenburg, A. Stolin, W. Szyman´ski, and L. Vainerman for discussions and advice. The second author is grateful to M.I.T. and P. Etingof for the kind hospitality during his visit.

Notes

or an irreducible strongly separable extension with Markov trace [10].