Hochschild Cohomology of Algebras in Monoidal Categories and Splitting Morphisms of Bialgebras

Details Hochschild Cohomology of Algebras in Monoidal Categories and Splitting Morphisms of Bialgebras Hochschild Cohomology of Algebras in Monoidal Categories and Splitting Morphisms of Bialgebras

2002-12-23

arxiv.org/abs/math/0212326v2

Mathematics

Quantum Algebra

amsart, 41 pages, 7 figures (EPS files), minor changes, typos corrected

Scientific paper

The main goal of this paper is to investigate the structure of Hopf algebras with the property that either its Jacobson radical is a Hopf ideal or its coradical is a subalgebra. In order to do that we define the Hochschild cohomology of an algebra in an abelian monoidal category. Then we characterize those algebras which have dimension less than or equal to 1 with respect to Hochschild cohomology. Now let us consider a Hopf algebra A such that its Jacobson radical J is a nilpotent Hopf ideal and H:=A/J is a semisimple algebra. By using our homological results, we prove that the canonical projection of A on H has a section which is an H-colinear algebra map. Furthermore, if H is cosemisimple too, then we can choose this section to be an (H,H)-bicolinear algebra morphism. This fact allows us to describe A as a `generalized bosonization' of a certain algebra R in the category of Yetter-Drinfeld modules over H. As an application we give a categorical proof of Radford's result about Hopf algebras with projections. We also consider the dual situation. In this case, many results that we obtain hold true for a large enough class of H-module coalgebras, where H is a cosemisimple Hopf algebra.

Affiliated with

Also associated with

No associations

Rating

Hochschild Cohomology of Algebras in Monoidal Categories and Splitting Morphisms of Bialgebras does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Hochschild Cohomology of Algebras in Monoidal Categories and Splitting Morphisms of Bialgebras, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Hochschild Cohomology of Algebras in Monoidal Categories and Splitting Morphisms of Bialgebras will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-74806