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Coring techniques and monoidal categories applied to Hopf algebras and their generalizations

maandag, 7 juni, 2010 - 09:45
Campus: Brussels Humanities, Sciences & Engineering campus
Faculteit: Science and Bio-engineering Sciences
Kris Janssen

In the past decades many generalizations of Hopf algebras have appeared in the literature. In this work we are particularly interested in multiplier Hopf algebras (MHAs) and Hopf group coalgebras, especially in their categorical behavior. In the second chapter we develop a theory of group corings, notably a Galois (descent) theory for such objects. As an application of the latter we obtain a Structure Theorem for relative Hopf group modules over a faithfully at Galois extension. We also discuss the relation between group corings and the dual notion of group-graded ring, and introduce and study strong group corings, dualizing strongly group-graded rings. In the third chapter we present two approaches to better understand MHAs from a categorical point of view. The first one makes use of the notion of a multiplier bialgebra, so that a MHA is a multiplier bialgebra along with some kind of antipode, as it is the case classically. The other one consists in the development of a general theory of so-called Kleisli-Hopf algebras. Examples of these are provided by a broad class of MHAs, as well as by Hopf group coalgebras. In the final chapter we generalize usual (co)actions of a Hopf algebra on an algebra to partial (co)actions, making use of coring techniques. Several examples are given, of which partial group actions are the most basic and motivational. The first chapter captures some (well-known) generalities on monoidal categories, Hopf algebras and corings.