Higher Frobenius-Schur Indicators for Semisimple Hopf Algebras in Positive Characteristic

Let [Formula: see text] be a semisimple Hopf algebra over an algebraically closed field [Formula: see text] of characteristic [Formula: see text]. We show that the antipode [Formula: see text] of [Formula: see text] satisfies the equality [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] is a nonzero integral of [Formula: see text]. The formula of [Formula: see text] enables us to define higher Frobenius-Schur indicators for the Hopf algebra [Formula: see text]. This generalizes the notion of higher Frobenius-Schur indicators from the case of characteristic 0 to the case of characteristic [Formula: see text]. These indicators defined here share some properties with the ones defined over a field of characteristic 0. In particular, all these indicators are gauge invariants for the tensor category [Formula: see text] of finite-dimensional representations of [Formula: see text].