Higher Frobenius-Schur indicators for semisimple Hopf algebras in positive characteristic

Let H be a semisimple Hopf algebra over an algebraically closed field k of characteristic p > dim k (H). We show that the antipode S of H satisfies the equality S (h) = uhu, where h ∈ H, u = S (Λ(2))Λ(1) and Λ is a nonzero integral of H. The formula of S 2 enables us to define higher FrobeniusSchur indicators for the Hopf algebra H. This generalizes the notions of higher Frobenius-Schur indicators from the case of characteristic 0 to the case of characteristic p > dim k (H). These indicators defined here share some properties with the ones defined over a field of characteristic 0. Especially, all these indicators are gauge invariants for the tensor category Rep(H) of finite dimensional representations of H.