A monoidal approach to splitting morphisms of bialgebras
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- by A. Ardizzoni, C. Menini and D. Ştefan PDF
- Trans. Amer. Math. Soc. 359 (2007), 991-1044 Request permission
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 $H$.References
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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. Ştefan
- Affiliation: Faculty of Mathematics, University of Bucharest, Strada Academiei 14, Bucharest, RO-70109, Romania
- Email: dstefan@al.math.unibuc.ro
- 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.
- © Copyright 2006 American Mathematical Society
- 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
- MathSciNet review: 2262840