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.
Highlights
A Hopf algebra H is called co-Frobenius if H is either left or right co-Frobenius as a coalgebra, i.e., if there exists a left or right H* monomorphism from H to H*
Among the properties of finite dimensional Hopf algebras that hold for all co-Frobenius Hopf algebras are the bijectivity of the antipode, a bijective correspondence between the group-like elements of the Hopf algebra and the one dimensional ideals of the dual algebra, the existence of a distinguished group-like element, and a reasonable theory of Galois extensions
The class of infinite dimensional co-Frobenius Hopf algebras includes cosemisimple Hopf algebras, such as the group algebra of an infinite group. Tensoring such a Hopf algebra H with a finite dimensional Hopf algebra K, yields an infinite dimensional Hopf algebra with non-zero integral obtained by tensoring the integrals of H and K
Summary
A Hopf algebra H is called co-Frobenius if H is either left or right co-Frobenius as a coalgebra, i.e., if there exists a left or right H* monomorphism from H to H*. 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).
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