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Journal of Operator Theory

Volume 46, Issue 3, Supplementary 2001  pp. 635-655.

Duality for actions of weak Kac algebras

Authors Dmitri Nikshych
Author institution: University of New Hampshire, Department of Mathematics, Kingsbury Hall, Durham, NH 03824, USA

Summary:  Weak Kac algebras generalize both finite dimensional Kac algebras and groupoid algebras. They naturally arise as symmetries of depth $2$ inclusions of II${}_1$ factors ([16]). We show that indecomposable weak Kac algebras are free over their counital subalgebras and prove a duality theorem for their actions. Using this result, for any biconnected weak Kac algebra we construct a minimal action on the hyperfinite II${}_1$ factor. The corresponding crossed product inclusion of II${}_1$ factors has depth 2 and an integer index. Its first relative commutant is, in general, non-trivial, so we derive some arithmetic properties of weak Kac algebras from considering reduced subfactors.

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