# 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

## Access Full Article

top## Abstract

top## How to cite

topBespalov, 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

top- [1] Yu. Bespalov, Crossed Modules and Quantum Groups in Braided Categories, to appear in Appl. Categorical Structures.
- [2] Yu. Bespalov and B. Drabant, Hopf (Bi-)Modules and Crossed Modules in Braided Monoidal Categories, preprint UVA-FWI 95-18 (1995).
- [3] Yu. Bespalov and B. Drabant, Differential Calculus in Braided Abelian Categories, in preparation.
- [4] T. Brzezinski, Remarks on Bicovariant Differential Calculi and Exterior Hopf Algebras, Lett. Math. Phys. 27, 287 (1993). Zbl0782.17009
- [5] B. Drabant, Braided Bosonization and Inhomogeneous Quantum Groups, Preprint UVA-FWI 94-28 (1994), to appear in Acta Math. Appl. Zbl0882.16023
- [6] P. Freyd and D. Yetter, Braided compact closed categories with applications to low dimensional topology, Adv. Math. 77, 156 (1989). Zbl0679.57003
- [7] D. Husemoller, Cyclic Homology, Tata Institute Lecture Notes on Mathematics 83, Springer (1991).
- [8] A.P. Isaev and A.A. Vladimirov, $G{L}_{q}\left(n\right)$-Covariant Braided Differential Bialgebras, Lett. Math. Phys. 33, 297 (1995).
- [9] A. Joyal and R. Street, Braided monoidal categories, Mathematics Reports 86008, Macquarie University (1986). Zbl0817.18007
- [10] S. Majid, Algebras and Hopf algebras in braided categories, Advances in Hopf Algebras, Lecture Notes in Pure and Appl. Math. 158, 55, Dekker (1994). Zbl0812.18004
- [11] S. Majid, Cross Products by Braided Groups and Bosonization, J. Algebra 163, 165 (1994). Zbl0807.16036
- [12] S. Majid, Transmutation theory and rank for quantum braided groups, Math. Proc. Camb. Phil. Soc. 113, 45 (1993). Zbl0781.17006
- [13] S. Majid, Free Braided Differential Operators, Braided Binomial Theorem, and the Braided Exponential Map, J. Math. Phys. 34, 4843 (1993). Zbl0807.16035
- [14] S. Mac Lane, Categories. For the Working Mathematician, GTM 5, Springer (1972).
- [15] S. Mac Lane, Natural associativity and commutativity, Rice University Studies 49, 28 (1963).
- [16] Yu. I. Manin, Notes on Quantum Groups and Quantum de Rham Complexes, J. Theor. and Math. Phys. 92, 425 (1992). G. Maltsiniotis, Le langage des espaces et des groupes quantiques, Commun. Math. Phys. 151, 275 (1993).
- [17] D.E. Radford, The Structure of Hopf Algebras with a Projection, J. Algebra 92, 322 (1985). Zbl0549.16003
- [18] D.E. Radford and J. Towber, Yetter-Drinfel'd categories associated to an arbitrary bialgebra, J. Pure Appl. Algebra 87, 259 (1993). Zbl0796.16033
- [19] M.E. Sweedler, Hopf algebras, W.A. Benjamin, New York (1969).
- [20] S.L. Woronowicz, Differential Calculus on Compact Matrix Pseudogroups (Quantum Groups), Commun. Math. Phys. 122, 125 (1989). Zbl0751.58042
- [21] D. Yetter, Quantum groups and representations of monoidal categories, Math. Proc. Camb. Phil. Soc. 108, 261 (1990). Zbl0712.17014

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.