On the Antipode of a Co-Frobenius (Co)Quasitriangular Hopf Algebra

Abstract

For H a quasitriangular Hopf algebra, S ², the square of the antipode is the inner automorphism induced by the Drinfeld element u, and S ⁴ is the inner automorphism induced by the grouplike element g = uS(u)⁻¹. For H finite dimensional, results of Drinfeld and Radford express g in terms of the modular elements of H. This note supplies another proof which replaces the requirement of finite dimensionality with existence of a nonzero integral for H in H*. Similar results hold for the infinite dimensional coquasitriangular case; here we supply some interesting examples.

Key Words:

• Coquasitriangular Hopf algebras
• Quasitriangular Hopf algebra

2000 Mathematics Subject Classification:

• 16W30
• 16W20

ACKNOWLEDGMENTS

The second author held a postdoctoral fellowship at Mount Allison University from 2005 to 2007 and would like to thank Mount Allison for their warm hospitality. Support for the first author's research and partial support for the postdoctoral position of the second author came from an NSERC Discovery Grant. The second author now holds research support from Grant 434/1.10.2007 of CNCSIS (ID 1005).

Notes

Communicated by M. Cohen.