Abstract

In the last decade various motivations coming from low dimensional quantum field theory, operator algebra, and Poisson geometry have lead to the introduction of a new notion of symmetry that generalizes both quantum groups and classical groupoid algebras. The slightly different definitions [18, 29, 12, 2, 16, 26, 3, 23, 9] all share in the property that a ”quantum groupoid ” A contains two antiisomorphic canonical subalgebras: The source subalgebra A R and the target subalgebra A L. They reduce to the scalars in the case of a Hopf algebra and they are the carrying space of the trivial representation in the monoidal category MA of A-modules. In the groupoid interpretation A L and A R are non-commutative analogues of the algebra of functions on the space of units. The various definitions of quantum groupoid differ in the size and in commutativity of these subalgebras. The most general among them is Lu’s Hopf algebroid [16], while the (C ∗)-weak Hopf algebra of [2, 3] captures the most general ”finite quantum groupoid ” (see [22] for a review) which has the extra beauty of being selfdual, just like a finite Abelian group or a finite dimensional Hopf algebra. Yamanouchi’s generalized Kac algebra

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