A monoidal approach to splitting morphisms of bialgebras

• Published in: Transactions of the American Mathematical Society Vol. 359; no. 3; pp. 991 - 1044
• Main Authors: Ardizzoni, A., Menini, C., Ştefan, D.
• Format: Journal Article
• Language: English
• Published: Providence, RI American Mathematical Society 01-03-2007
• Subjects: Adjoints; Algebra; Associative rings and algebras; Braiding; Exact sciences and technology; Functors; Linear transformations; Mathematical objects; Mathematical theorems; Mathematics; Morphisms; Sciences and techniques of general use; Subalgebras; bialgebras; smash (co)products; monoidal categories; Hopf algebras; Coalgebra; Splitting; Bialgebra; Ideal; Semisimple algebra; Morphism; Monoidal category; Hopf algebra
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-06-03902-X

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. 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. 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. Let A be a bialgebra such that its coradical is a Hopf sub-bialgebra with antipode. Then there is a retraction of the canonical injection of H into A which is an H--linear coalgebra morphism. Furthermore, if H is semisimple too, then we can choose this retraction to be an (H,H)--bilinear coalgebra morphism. Then, also in this case, we can describe A as a `generalized bosonization' of a certain coalgebra R in the category of Yetter--Drinfeld modules over H.