THE BRAIDED STRUCTURES FOR T-SMASH PRODUCT HOPF ALGEBRAS

Yıl 2007, Cilt: 1 Sayı: 1, 30 - 45, 01.06.2007

Zhengming Jiao Robert Wisbauer

Öz

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.

Anahtar Kelimeler

Braided Hopf algebra, T-smash product Hopf algebra

Kaynakça

• 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: [e-mail] Robert Wisbauer
• Department of Mathematics, Heinrich University D¨usseldorf, Germany, e-mail: [e-mail]