Finite dimensional graded simple algebras

Abstract

Let $R$ be a finite-dimensional algebra over an algebraically closed field $F$ graded by an arbitrary group $G$. We prove that $R$ is a graded division algebra if and only if it is isomorphic to a twisted group algebra of some finite subgroup of $G$. If the characteristic of $F$ is zero or $\mathrm{char}F$ does not divide the order of any finite subgroup of $G$ then we prove that $R$ is graded simple if and only if it is a matrix algebra over a finite-dimensional graded division algebra.