Radford's Formula for Bifrobenius Algebras and Applications

Abstract

Abstract. In a biFrobenius algebra H, in particular in the case that H is a finite dimensional Hopf algebra, the antipode S : H → H can be decomposed as S =tc•c φ where c φ : H → H * and tc : H * → H are the Frobenius and coFrobenius isomorphisms. We use this decomposition to present an easy proof of Radford's formula for S 4 . Then, in the case that the map S satisfies the additional condition that S ⋆ id = id ⋆ S = uε, we prove the trace formula: tr(S 2 ) = ε(t)φ(1) . We finish by applying the above results to study the semisimplicity and cosemisimplicity of H.