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THE BRAIDED STRUCTURES FOR T-SMASH PRODUCT HOPF ALGEBRAS

Year 2007, Volume: 1 Issue: 1, 30 - 45, 01.06.2007

Zhengming Jiao Robert Wisbauer

Abstract

As a dual concept of quasitriangular bialgebra, braided bialgebras
were introduced by Larson and Towber. The braided structures of ω-smash
coproduct Hopf algebras have been investigated recently by the authors. Here
we study the braided structures of T-smash product Hopf algebras BonT H as
constructed by Caenepeel, Ion, Militaru and Zhu. Necessary and sufficient
conditions for T-smash product Hopf algebras to be braided Hopf algebras
are given in terms of properties of their components. We apply our results to
discuss some special cases. In particular, braided structures of the Drinfeld
double D(H) and of H4onT RZ2 = H4 ∗RZ2 (skew-group ring) are constructed.

Keywords

Braided Hopf algebra T-smash product Hopf algebra

References

  • T. Brzezi´nski and R. Wisbauer, Corings and Comodules, Cambridge Univ. Press, 2003.
  • S. Caenepeel, B. Ion, G. Militaru and Shenglin Zhu, The factorization problem and the smash biproduct of algebras and coalgebras, Algebras and Repr. Theory (2000), 19-42.
  • Y. Doi, Braided bialgebras and quadratic bialgebras, Comm. Algebra 21(5) (1993), 1731-1749.
  • Y. Doi and M. Takeuchi, Multiplication alteration by two-cocycles, Comm. Algebra 22(14) (1994), 5715-5732.
  • V.G. Drinfeld, Quantum groups, in: Proceedings of the Int. Congress of Math., Berkeley (1987), 798-820.
  • Z. Jiao and R. Wisbauer, The braided structures for ω-smash coproduct Hopf algebras, J. Algebra 287 (2005), 474-495.
  • R. Larson and J. Towber, Two dual classes of bialgebras related to the concepts of “quantum groups” and “quantum Lie algebras”, Comm. Algebra 19 (1991), 3345.
  • S. Majid, Foundations of Quantum Group Theory, Cambridge Univ. Press, R.K. Molnar, Semi-direct products of Hopf algebras, J. Algebra 47 (1977), 29
  • D.E. Radford, Minimal quasitriangular Hopf algebra, J. Algebra, 157 (1993), 235.
  • D.E. Radford, On Kauffman’s knot invariants arising from finite-dimensional Hopf algebras, in: Advances in Hopf algebras, LNPAM 158, Dekker, New York, , pp. 205-266. M.E. Sweedler, Hopf Algebras, Benjamin, New York, 1969.
  • S. Wang and J. Li, On twisted smash products for bimodule algebras and the Drinfeld double, Comm. Algebra 26(8) (1998), 2435-2444. Zhengming Jiao
  • Department of Mathematics, Henan Normal University Xinxiang, Henan 453007, PR China, e-mail: zmjiao@371.net Robert Wisbauer
  • Department of Mathematics, Heinrich University D¨usseldorf, Germany, e-mail: wisbauer@math.uni-duesseldorf.de

Year 2007, Volume: 1 Issue: 1, 30 - 45, 01.06.2007

Zhengming Jiao Robert Wisbauer

Abstract

References

  • T. Brzezi´nski and R. Wisbauer, Corings and Comodules, Cambridge Univ. Press, 2003.
  • S. Caenepeel, B. Ion, G. Militaru and Shenglin Zhu, The factorization problem and the smash biproduct of algebras and coalgebras, Algebras and Repr. Theory (2000), 19-42.
  • Y. Doi, Braided bialgebras and quadratic bialgebras, Comm. Algebra 21(5) (1993), 1731-1749.
  • Y. Doi and M. Takeuchi, Multiplication alteration by two-cocycles, Comm. Algebra 22(14) (1994), 5715-5732.
  • V.G. Drinfeld, Quantum groups, in: Proceedings of the Int. Congress of Math., Berkeley (1987), 798-820.
  • Z. Jiao and R. Wisbauer, The braided structures for ω-smash coproduct Hopf algebras, J. Algebra 287 (2005), 474-495.
  • R. Larson and J. Towber, Two dual classes of bialgebras related to the concepts of “quantum groups” and “quantum Lie algebras”, Comm. Algebra 19 (1991), 3345.
  • S. Majid, Foundations of Quantum Group Theory, Cambridge Univ. Press, R.K. Molnar, Semi-direct products of Hopf algebras, J. Algebra 47 (1977), 29
  • D.E. Radford, Minimal quasitriangular Hopf algebra, J. Algebra, 157 (1993), 235.
  • D.E. Radford, On Kauffman’s knot invariants arising from finite-dimensional Hopf algebras, in: Advances in Hopf algebras, LNPAM 158, Dekker, New York, , pp. 205-266. M.E. Sweedler, Hopf Algebras, Benjamin, New York, 1969.
  • S. Wang and J. Li, On twisted smash products for bimodule algebras and the Drinfeld double, Comm. Algebra 26(8) (1998), 2435-2444. Zhengming Jiao
  • Department of Mathematics, Henan Normal University Xinxiang, Henan 453007, PR China, e-mail: zmjiao@371.net Robert Wisbauer
  • Department of Mathematics, Heinrich University D¨usseldorf, Germany, e-mail: wisbauer@math.uni-duesseldorf.de
There are 13 citations in total.

Details

Other ID JA59HS74GF
Journal Section Articles
Authors

Zhengming Jiao This is me

Robert Wisbauer This is me

Publication Date June 1, 2007
Published in Issue Year 2007 Volume: 1 Issue: 1

Cite

APA Jiao, Z., & Wisbauer, R. (2007). THE BRAIDED STRUCTURES FOR T-SMASH PRODUCT HOPF ALGEBRAS. International Electronic Journal of Algebra, 1(1), 30-45.
AMA Jiao Z, Wisbauer R. THE BRAIDED STRUCTURES FOR T-SMASH PRODUCT HOPF ALGEBRAS. IEJA. June 2007;1(1):30-45.
Chicago Jiao, Zhengming, and Robert Wisbauer. “THE BRAIDED STRUCTURES FOR T-SMASH PRODUCT HOPF ALGEBRAS”. International Electronic Journal of Algebra 1, no. 1 (June 2007): 30-45.
EndNote Jiao Z, Wisbauer R (June 1, 2007) THE BRAIDED STRUCTURES FOR T-SMASH PRODUCT HOPF ALGEBRAS. International Electronic Journal of Algebra 1 1 30–45.
IEEE Z. Jiao and R. Wisbauer, “THE BRAIDED STRUCTURES FOR T-SMASH PRODUCT HOPF ALGEBRAS”, IEJA, vol. 1, no. 1, pp. 30–45, 2007.
ISNAD Jiao, Zhengming - Wisbauer, Robert. “THE BRAIDED STRUCTURES FOR T-SMASH PRODUCT HOPF ALGEBRAS”. International Electronic Journal of Algebra 1/1 (June 2007), 30-45.
JAMA Jiao Z, Wisbauer R. THE BRAIDED STRUCTURES FOR T-SMASH PRODUCT HOPF ALGEBRAS. IEJA. 2007;1:30–45.
MLA Jiao, Zhengming and Robert Wisbauer. “THE BRAIDED STRUCTURES FOR T-SMASH PRODUCT HOPF ALGEBRAS”. International Electronic Journal of Algebra, vol. 1, no. 1, 2007, pp. 30-45.
Vancouver Jiao Z, Wisbauer R. THE BRAIDED STRUCTURES FOR T-SMASH PRODUCT HOPF ALGEBRAS. IEJA. 2007;1(1):30-45.