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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 Citation[10].