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

TL;DR

This paper develops Hochschild cohomology for algebras in monoidal categories to analyze the structure of Hopf algebras with specific radical and coradical properties, leading to new decomposition results.

Contribution

It introduces Hochschild cohomology in monoidal categories and applies it to characterize and decompose Hopf algebras with nilpotent radicals and semisimple quotients.

Findings

Canonical projection admits an H-colinear algebra section.

A Hopf algebra can be described as a generalized bosonization.

Provides a categorical proof of Radford's theorem.

Abstract

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.