Ribbon Element on Co-Frobenius Quasitriangular Hopf Algebras

Guohua Liu
DOI: 10.4236/am.2010.13028                

Abstract

Let (H, R) be a co-Frobenius quasitriangular Hopf algebra with antipode S. Denote the set of group-like elements in H by G (H). In this paper, we find a necessary and sufficient condition for (H, R) to have a ribbon element. The condition gives a connection with the order of G (H) and the order of S2.

Keywords

Co-Frobenius Hopf Algebra, Ribbon Element

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	B. I. P. Lin, “Semiperfect Coalgebras,” Journal of Algebra, Vol. 49, No. 2, 1977, pp. 357-373.
[2]	Y. Doi, “Homological Coalgebra,” Journal of the Mathematical Society of Japan, Vol. 33, No. 1, 1981, pp. 31-50.
[3]	A. van Daele, “An Algebraic Framework for Group Duality,” Advances in Mathematics, Vol. 140, No. 2, 1998, pp. 323-366.
[4]	N. Y. Reshetikhin and V. G. Turaev, “Ribbon Graphs and their Invariants Derived from Quantum Groups,” Communications in Mathematical Physics, Vol. 127, No. 1, 1990, pp. 1-26.
[5]	N. Y. Reshetikhin and V. G. Turaev, “Invariants of 3-Manifolds via Link Polynomials and Quantum Groups,” Invented Mathematics, Vol. 103, No. 3, 1991, pp. 547-597.
[6]	R. Kirby and P. Melvin, “The 3-Manifolds Invariants of Witten and Reshetikhin-Turaev for sl (2, C),” Invented Mathematics, Vol. 105, No. 3, 1991, pp. 473-545.
[7]	M. Beattie, D. Bulacu and B. Torrecillas, “Radford’s s4 Formula for Co-Frobenius Hopf Algebras,” Journal of Algebra, Vol. 307, No. 1, 2007, pp. 330-342.
[8]	M. Beattie, S. Dascalescu, L. Grunenfelder and C. Nastasescu, “Finiteness Conditions, Co-Frobenius Hopf Algebras and Quantum Groups,” Journal of Algebra, Vol. 200, No. 1, 1998, pp. 312- 333.
[9]	M. Beattie, S. Dascalescu and S. Raianu, “Galois Extensions for Co-Frobenius Hopf Algebras,” Journal of Algebra, Vol. 198, No. 1, 1997, pp. 164-183.
[10]	L. H. Kauffman and D. E. Radford, “A Necessary and Sufficient Condition for a Finite–Dim–Ensional Drinfeld Double to be a Ribbon Hopf Algebras,” Journal of Algebra, Vol. 159, No. 1, 1993, pp. 98-114.
[11]	V. G. Drinfeld, “On Almost Cocommutative Hopf Algebras,” Leningrad Mathematical Journal, Vol. 1, No. 2, 1990, pp. 321-342.
[12]	D. E. Radford, “On the Antipode of a Quasitriangular Hopf Algebras,” Journal of Algebra, Vol. 151, No. 1, 1992, pp. 1-11.
[13]	M. Beattie and D. Bulacu, “On The Antipode of a Co-Fro- benius (Co) Quasitriangular Hopf Algebras,” Communications in Algebra, Vol. 37, No. 9, 2009, pp. 2981-2993.
[14]	N. Y. Reshetikhin and V. G. Turaev, “Invariants of 3-Mani- folds via Link Polymonials and Quantum Groups,” Com- munications in Mathematical Physics, Vol. 127, No. 1, 1990, pp. 7-26.