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

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 such that its Jacobson radical is a nilpotent Hopf ideal and is a semisimple algebra. We prove that the canonical projection of on has a section which is an -colinear algebra map. Furthermore, if is cosemisimple too, then we can choose this section to be an -bicolinear algebra morphism. This fact allows us to describe as a `generalized bosonization' of a certain algebra in the category of Yetter-Drinfeld modules over . As an application we give a categorical proof of Radford's result about Hopf algebras with projections. We also consider the dual situation. Let be a bialgebra such that its coradical is a Hopf sub-bialgebra with antipode. Then there is a retraction of the canonical injection of into which is an -linear coalgebra morphism. Furthermore, if is semisimple too, then we can choose this retraction to be an -bilinear coalgebra morphism. Then, also in this case, we can describe as a `generalized bosonization' of a certain coalgebra in the category of Yetter-Drinfeld modules over .

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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