Hochschild Cohomology of Algebras in Monoidal Categories and Splitting Morphisms of Bilgebras
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چکیده
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.
منابع مشابه
Hochschild Cohomology of Algebras in Monoidal Categories and Splitting Morphisms of Bialgebras
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I work on the cohomology, structure, and representations of various types of rings, such as Hopf algebras and group-graded algebras. My research program involves collaborations with many mathematicians, including work with postdocs and graduate students. Below is a summary of some of my past and ongoing research projects, which fall loosely into three categories: Hochschild cohomology and defor...
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تاریخ انتشار 2003