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A monoidal approach to splitting morphisms of bialgebras


Authors: A. Ardizzoni, C. Menini and D. Stefan
Journal: Trans. Amer. Math. Soc. 359 (2007), 991-1044
MSC (2000): Primary 16W30; Secondary 16S40
DOI: https://doi.org/10.1090/S0002-9947-06-03902-X
Published electronically: October 17, 2006
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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. 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$.


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Additional Information

A. Ardizzoni
Affiliation: Department of Mathematics, University of Ferrara, Via Machiavelli 35, Ferrara, I-44100, Italy
Email: alessandro.ardizzoni@unife.it

C. Menini
Affiliation: Department of Mathematics, University of Ferrara, Via Machiavelli 35, I-44100, Ferrara, Italy
Email: men@dns.unife.it

D. Stefan
Affiliation: Faculty of Mathematics, University of Bucharest, Strada Academiei 14, Bucharest, RO-70109, Romania
Email: dstefan@al.math.unibuc.ro

DOI: https://doi.org/10.1090/S0002-9947-06-03902-X
Keywords: Hopf algebras, bialgebras, smash (co)products, monoidal categories
Received by editor(s): July 1, 2004
Received by editor(s) in revised form: November 3, 2004, and November 17, 2004
Published electronically: October 17, 2006
Additional Notes: This paper was written while the first two authors were members of G.N.S.A.G.A. with partial financial support from M.I.U.R. The third author was partially supported by I.N.D.A.M., while he was a visiting professor at the University of Ferrara.
Article copyright: © Copyright 2006 American Mathematical Society

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