TWISTED SMASH COPRODUCTS IN BRAIDED MONOIDAL CATEGORIES

Year 2010, Volume: 7 Issue: 7, 12 - 33, 01.06.2010

Wenzheng Zhao Shengjie Gao Tianshui Ma

Abstract

Let A ×r H be a twisted smash coproduct for a bicomodule coalgebra A by a Hopf algebra H in a braided monoidal category. The smash coproduct in a braided monoidal category is the special case of A ×r H. Moreover, we find a necessary and sufficient condition for A ×r H to be a bialgebra.

Keywords

braided monoidal categories, twisted smash coproducts, braided Hopf algebras, braided bicomodule coalgebras

References

• P. Deligne and J. S. Milne, Tannakian Categories, Lecture Notes in Math., Vol. 900, Springer, 1982.
• P. Freyd and D. Yetter, Braided compact closed categories with applications to low dimen- sional topology, Adv. Math., 77 (1989), 156-182.
• R. G. Heyneman and M. E. Sweedler, Affine Hopf algebra, I., J. Algebra, 13 (1969), 192
• A. Joyal and R. Street, Braided monoidal categories, Adv. Math., 102 (1993), 20-78.
• C. Kassel, Quantum Groups, Springer-Verlag, New York, 1995.
• J. Q. Li and Y. H. Xu, Smash coproduct and braided groups in braided monoidal category, Science in China, Series A, (1997) 1121-1128.
• V. Lyubashenko, Tangles and Hopf algebras in braided tensor categories, J. Pure Appl. Algebra, 98 (1995), 245-278.
• S. Mac Lane, Categories for the Working Mathematician, GTM Vol. 5, Springer, Berlin/New York, 1971.
• S. Majid, Algebras and Hopf algebras in braided categories, Advances in Hopf algebras (Chicago, IL, 1992), 55-105, Lecture Notes in Pure and Appl. Math., 158, Dekker, New York, 1994.
• S. Majid, Foundations of Quantum Group Theory, Cambridge University Press, Cam- bridge, 1995.
• S. Majid and M. J. Rodr´ıguez-Plaza, Random walk and the heat equation on superspace and anyspace, J. Math. Phys., 35(7) (1994), 3753-3760.
• S. Majid, Beyond supersymmetry and quantum symmetry(an introduction to braided groups and braided matries), (Proceeding of the 5th Nakai workshop, Tianjin, China), Quantum Groups, Integrable Statistical Models and Knot Theory, 1992, edited by M. L. Ge and H.J.
• De Vega, World Scientific, Singapore, 1993, 231-282.
• S. Majid, Braided groups and algebraic quantum field theories, Lett. Math. Phys., 22 (1991), 175.
• S. Majid, Some physical applications of category theory, In Proceeding of the 19th DGM, Rapallo, 1990, Lecture Notes in physics, Vol. 375(Springer, Berlin, 1991), 131-142.
• S. Majid, Braided groups, J. Pure Appl. Algebra, 86 (1993), 187-221.
• S. Majid, Transmutation theory and rank for quantum braided groups, Math. Proc. Camb. Phil. Soc., 113 (1993), 45-70.
• S. Majid, More examples of bicrossproduct and double cross product Hopf algebras, Israel J. Math., 72 (1990), 133-148.
• S. Majid, Cross products by braided groups and Bosonization, J. Algebra, 163 (1994), 165
• R. K. Molnar, Semi-direct products of Hopf algebras, J. Algebra, 47 (1977), 29-51.
• S. H. Wang and J. Q. Li, On twisted smash product for bimodule algebras and the Drinfeld double, Comm. Algebra, 26(8) (1998), 2435-2444. Wenzheng Zhao
• College of Mathematics and Information Science Henan Normal University Xinxiang, P. R. China e-mail: [e-mail] Shengjie Gao College of Mathematics and Information Science Henan Normal University Xinxiang, P. R. China e-mail: [e-mail] Tianshui Ma College of Mathematics and Information Science Henan Normal University Xinxiang, P. R. China e-mail: [e-mail]