Alessandro Ardizzoni

Separable Functors and Formal Smoothness

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.The notion of a formally smooth bimodule is introduced and its basic properties are analyzed. In particular it is proven that a

Cocycle Deformations for Hopf Algebras with a Coalgebra Projection

B

BRAIDED BIALGEBRAS OF TYPE ONE

-

The Category of Modules over a Monoidal Category: Abelian or not?

A

Cotensor Coalgebras in Monoidal Categories

bimodule

Quadratic Lie Algebras

M

Weak projections onto a braided Hopf algebra

which is a generator left

Associated Graded Algebras and Coalgebras

B

A Milnor–Moore type theorem for primitively generated braided bialgebras

-module is formally smooth if and only if the

GAUGE DEFORMATIONS FOR HOPF ALGEBRAS WITH THE DUAL CHEVALLEY PROPERTY

M