Hochschild Cohomology of Algebras in Monoidal Categories and Splitting Morphisms of Bialgebras
نویسنده
چکیده
Let (M,⊗,1) be an abelian monoidal category. In order to investigate the structure of Hopf algebras with Chevalley property (i.e. Hopf algebras having the coradical a Hopf subalgebra) we define the Hochschild cohomology of an algebra A in M. Then we characterize those algebras A which have dimension less than or equal to 1 with respect to Hochschild cohomology. We use these results to prove that a Hopf algebra A with the Chevalley property has a projection π : A → H which is a right H– colinear map coalgebra map, where H is the coradical of A (this result has been also proved by A. Masuoka). As a consequence we deduce a formula for the first coradical filtration of A. If in addition H is semisimple then this projection can be chosen such that it is a morphism of (H,H)– bicomodules, fact that allows us to describe A as a ‘generalized bosonisation’ of a certain object R by H (we would like to mention that these results hold true in a more general context, but for shortness we state them here in this particular form). The dual situation is also considered, and as an application we give a categorical proof of Radford’s result about Hopf algebras with projections.
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