Radford's Formula for Bifrobenius Algebras and Applications

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 𝒮⁴. Then, in the case that the map 𝒮 satisfies the additional condition that 𝒮  id = id  𝒮 = u ?, we prove the trace formula tr(𝒮²) = ?(t)φ(1). We finish by applying the above results to study the semisimplicity and cosemisimplicity of H.