2021/12/31 by Wang, Zhihua, Liu, Gongxiang, Li, Libin
#16T05 #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2112.15264
Let H be a semisimple Hopf algebra over an algebraically closed field \mathbbmk of characteristic p>dim_\mathbbmk(H)1/2. We show that the antipode S of H satisfies the equality S2(h)=uhu-1, where h∈ H, u=S(Λ(2))Λ(1) and Λ is a nonzero integral of H. The formula of S2 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>dim_\mathbbmk(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.