Derivation over twisted group ring and its applications
Alvaro Otero Sanchez
TL;DR
This work addresses the classification of derivations on twisted group algebras $R^{\alpha}G$, extending the Ferrero–Giambruno–Polcino Milies framework to the twisted setting and linking derivations to Hochschild cohomology $HH^1$.The authors develop a generator-relations extension criterion that yields a closed-form, unique derivation extension $f^*$ from a map on a generating set, and they prove a twisted-version of the FGP theorem showing innerness under suitable hypotheses.They apply the theory to finite abelian groups and to dihedral groups, deriving explicit dimension counts for $Der(\mathbb{F}_q^{\alpha}G)$ and exact formulas for $HH^1(\mathbb{F}_q^{\alpha}D_{2n})$ across parity and cocycle cases, including nontrivial centers and explicit bases in several examples.The results illuminate how cocycle twists influence derivations and Hochschild cohomology, with potential implications for representation theory, coding theory via twisted group rings, and the structural study of noncommutative algebras.
Abstract
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.