Separable Functors and Formal Smoothness

Abstract

AbstractThe 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 ofad-(co)invariant integrals for a Hopf algebraHis equivalent to the separability of some forgetful functors. In the finite dimensional case, this is also equivalent to the separability of the Drinfeld DoubleD(H)overH. Hopf algebras which are formally smooth as (co)algebras are characterized. We prove that if π :E→His a bialgebra surjection with nilpotent kernel such thatHis a Hopf algebra which is formally smooth as aK-algebra, then π has a section which is a rightH-colinear algebra homomorphism. Moreover, ifHis also endowed with anad-invariant integral, then this section can be chosen to beH-bicolinear. We also deal with the dual case.