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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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 $H$.
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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