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Regular multiplier Hopf algebroids. Basic theory and examples

Thomas Timmermann, Alfons van Daele
published 2013-07-31

Multiplier Hopf algebroids are algebraic versions of quantum groupoids. They generalize Hopf bialgebroids to the non-unital case and need no separability assumption on the base as weak (multiplier) Hopf algebras do. Examples include function algebras on etale groupoids, which are Hopf bialgebroids only in the compact and weak multiplier Hopf algebras only in the discrete case. The main structure maps of a multiplier Hopf algebroid are a left and a right comultiplication which take values in restricted multiplier algebras of Takeuchi products. Equivalently, the comultiplication can be described in terms of four canonical maps satisfying a few key relations. The main result of this article, proved in a very transparent diagrammatic way, is that bijectivity of these maps is equivalent to the existence of counits and an invertible antipode. This extends corresponding results for Hopf algebroids and regular (weak) multiplier Hopf algebras. Regularity refers to the property that the co-opposite is a multiplier Hopf algebroid again, which is equivalent to invertibility of the antipode.

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