Forms of certain hopf algebras

Abstract

Let Ψ be a field, G a finite group of automorphisms of Ψ, and Φ the fixed field of G. Let H be a Hopf algebra over Ψ. For g ∈ G we define a Hopf algebra H_g which has the same underlying vector space as H and modified operations and show that the tensor product (over Ψ) ⊗_{g ∈ G} H_g has a Φ-form. As a consequence we see that if n>0 is an integer and Φ is a field of characteristic zero or p>0 with (n,p)=1, then there is a finite dimensional Hopf algebra over Φ with antipode of order 2n.