The Larson-Sweedler theorem says that a finite-dimensional bialgebra with a faithful integral is a Hopf algebra [L-S]. The result has been generalized to finite-dimensional weak Hopf algebras by Vecsernyés [Ve]. In this paper, we show that the result is still true for weak multiplier Hopf algebras.
In this series of papers, we develop the theory of a class of locally compact quantum groupoids, which is motivated by the purely algebraic notion of weak multiplier Hopf algebras. In this Part I, we provide motivation and formulate the definition in the C * -algebra framework. Existence of a certain canonical idempotent element is required and it plays a fundamental role, including the establishment of the coassociativity of the comultiplication. This class contains locally compact quantum groups as a subclass.
Abstract. As a generalization of the linear Poisson bracket on the dual space of a Lie algebra, we introduce certain non-linear Poisson brackets which are "cocycle perturbations" of the linear Poisson bracket. We show that these special Poisson brackets are equivalent to Poisson brackets of central extension type, which resemble the central extensions of an ordinary Lie bracket via Lie algebra cocycles. We are able to formulate (strict) deformation quantizations of these Poisson brackets by means of twisted group C * -algebras. We also indicate that these deformation quantizations can be used to construct some specific non-compact quantum groups.
This is Part II in our multi-part series of papers developing the theory of a subclass of locally compact quantum groupoids (quantum groupoids of separable type), based on the purely algebraic notion of weak multiplier Hopf algebras. The definition was given in Part I. The existence of a certain canonical idempotent element E plays a central role. In this Part II, we develop the main theory, discussing the structure of our quantum groupoids. We will construct from the defining axioms the right/left regular representations and the antipode map.2010 Mathematics Subject Classification. 46L65, 46L51, 81R50, 16T05, 22A22.
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