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Separable Functors and Formal Smoothness

Published online by Cambridge University Press:  21 December 2007

Alessandro Ardizzoni
Alessandro Ardizzoni
Affiliation:
[email protected] of Ferrara, Department of Mathematics, Via Machiavelli 35, Ferrara, I-44100, Italy
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Abstract

The natural problem we approach in the present paper is to show how the notion of formally smooth (co)algebra inside monoidal categories can substitute that of (co)separable (co)algebra in the study of splitting bialgebra homomorphisms. This is performed investigating the relation between formal smoothness and separability of certain functors and led to other results related to Hopf algebra theory. Between them we prove that the existence of ad-(co)invariant integrals for a Hopf algebra H is equivalent to the separability of some forgetful functors. In the finite dimensional case, this is also equivalent to the separability of the Drinfeld Double D(H) over H. Hopf algebras which are formally smooth as (co)algebras are characterized. We prove that if π : E → H is a bialgebra surjection with nilpotent kernel such that H is a Hopf algebra which is formally smooth as a K-algebra, then π has a section which is a right H-colinear algebra homomorphism. Moreover, if H is also endowed with an ad-invariant integral, then this section can be chosen to be H-bicolinear. We also deal with the dual case.

Keywords

Monoidal categoriesHopf algebrasseparable and formally smooth algebrasseparable functorsad-invariant integrals
Type
Research Article
Information
Journal of K-Theory , Volume 1 , Issue 3 , June 2008 , pp. 535 - 582
Copyright
Copyright © ISOPP 2008

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