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>_k(H)^1/2. We show that the antipode S of H satisfies the equality S^2(h)=𝐮h𝐮^-1, 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 Frobenius-Schur 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>_k(H)^1/2. 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.