Derivation over twisted group ring and its applications

This paper classifies the derivations of twisted group algebras in terms of the generators and defining relations of the group. In particular, we generalize some know results over group algebras to the case of twisted group algebras. We will give necessary and sufficient condition for a map defined over the generators to be extended to a derivation over a twisted group ring. We present how this results can be use in case of an abelian group or dihedral group. We also present how this results can be use to compute the first Hochschild cohomology group of twisted group algebras.