Bicovariant differential calculi and cross products on braided Hopf algebras

Yuri Bespalov; Bernhard Drabant

Banach Center Publications (1997)

  • Volume: 40, Issue: 1, page 79-90
  • ISSN: 0137-6934

Abstract

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In a braided monoidal category C we consider Hopf bimodules and crossed modules over a braided Hopf algebra H. We show that both categories are equivalent. It is discussed that the category of Hopf bimodule bialgebras coincides up to isomorphism with the category of bialgebra projections over H. Using these results we generalize the Radford-Majid criterion and show that bialgebra cross products over the Hopf algebra H are precisely described by H-crossed module bialgebras. In specific braided monoidal abelian categories we define (bicovariant) braided differential calculi over H and apply the results on Hopf bimodules to construct a higher order bicovariant differential calculus over H out of any first order bicovariant differential calculus over H. This object is shown to be a bialgebra with universal properties.

How to cite

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Bespalov, Yuri, and Drabant, Bernhard. "Bicovariant differential calculi and cross products on braided Hopf algebras." Banach Center Publications 40.1 (1997): 79-90. <http://eudml.org/doc/252253>.

@article{Bespalov1997,
abstract = {In a braided monoidal category C we consider Hopf bimodules and crossed modules over a braided Hopf algebra H. We show that both categories are equivalent. It is discussed that the category of Hopf bimodule bialgebras coincides up to isomorphism with the category of bialgebra projections over H. Using these results we generalize the Radford-Majid criterion and show that bialgebra cross products over the Hopf algebra H are precisely described by H-crossed module bialgebras. In specific braided monoidal abelian categories we define (bicovariant) braided differential calculi over H and apply the results on Hopf bimodules to construct a higher order bicovariant differential calculus over H out of any first order bicovariant differential calculus over H. This object is shown to be a bialgebra with universal properties.},
author = {Bespalov, Yuri, Drabant, Bernhard},
journal = {Banach Center Publications},
keywords = {braided monoidal categories; Hopf bimodules; braided Hopf algebras; categories of Hopf bimodule bialgebras; bialgebra cross products; braided monoidal Abelian categories; braided differential calculi},
language = {eng},
number = {1},
pages = {79-90},
title = {Bicovariant differential calculi and cross products on braided Hopf algebras},
url = {http://eudml.org/doc/252253},
volume = {40},
year = {1997},
}

TY - JOUR
AU - Bespalov, Yuri
AU - Drabant, Bernhard
TI - Bicovariant differential calculi and cross products on braided Hopf algebras
JO - Banach Center Publications
PY - 1997
VL - 40
IS - 1
SP - 79
EP - 90
AB - In a braided monoidal category C we consider Hopf bimodules and crossed modules over a braided Hopf algebra H. We show that both categories are equivalent. It is discussed that the category of Hopf bimodule bialgebras coincides up to isomorphism with the category of bialgebra projections over H. Using these results we generalize the Radford-Majid criterion and show that bialgebra cross products over the Hopf algebra H are precisely described by H-crossed module bialgebras. In specific braided monoidal abelian categories we define (bicovariant) braided differential calculi over H and apply the results on Hopf bimodules to construct a higher order bicovariant differential calculus over H out of any first order bicovariant differential calculus over H. This object is shown to be a bialgebra with universal properties.
LA - eng
KW - braided monoidal categories; Hopf bimodules; braided Hopf algebras; categories of Hopf bimodule bialgebras; bialgebra cross products; braided monoidal Abelian categories; braided differential calculi
UR - http://eudml.org/doc/252253
ER -

References

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  1. [1] Yu. Bespalov, Crossed Modules and Quantum Groups in Braided Categories, to appear in Appl. Categorical Structures. 
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  3. [3] Yu. Bespalov and B. Drabant, Differential Calculus in Braided Abelian Categories, in preparation. 
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  17. [17] D.E. Radford, The Structure of Hopf Algebras with a Projection, J. Algebra 92, 322 (1985). Zbl0549.16003
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