We introduce Brauer characters for representations of the bismash products of groups in characteristic p > 0, p not 2 and study their properties analogous to the classical case of finite groups. We then use our results to extend to bismash products a
theorem of Thompson on lifting Frobenius-Schur indicators from characteristic p to characteristic 0
In this paper we show that a certain algebra being a comodule algebra over the Taft Hopf algebra of dimension $n^2$ is equivalent to a set of identities related to the $q$-binomial coefficient, when $q$ is a primitive $n^{th}$ root of 1. We then give a direct combinatorial proof of these identities.
We introduce two kinds of gauge invariants for any finite-dimensional Hopf algebra H. When H is semisimple over C, these invariants are respectively, the trace of the map induced by the antipode on the endomorphism ring of a self-dual simple module,
and the higher Frobenius-Schur indicators of the regular representation. We further study the values of these higher indicators in the context of complex semisimple quasi-Hopf algebras H. We prove that these indicators are non-negative provided the module category over H is modular, and that for a prime p, the p-th indicator is equal to 1 if, and only if, p is a factor of dim H. As an application, we show the existence of a non-trivial self-dual simple H-module with bounded dimension which is determined by the value of the second indicator.
In this paper we consider gradings by a finite abelian group $G$ on the Lie algebra $mathfrak{sl}_n(F)$ over an algebraically closed field $F$ of characteristic different from 2 and not dividing $n$.
