Grading Structure for Derivations of Group Algebras

TL;DR

This paper introduces a method to impose a graded algebra structure on the derivation algebra of a group algebra using the derived group as the grading group, extending to outer derivations.

Contribution

It provides a novel approach to grading derivation algebras of group algebras based on the derived group, including outer derivations, for non-perfect groups.

Findings

Derived group used as grading group for derivation algebra

Established graded algebra structure for outer derivations

Non-trivial grading exists for all non-perfect groups

Abstract

In this paper we give a way of equipping the derivation algebra of a group algebra with the structure of a graded algebra. The derived group is used as the grading group. For the proof, the identification of the derivation with the characters of the adjoint action groupoid is used. These results also allow us to obtain the analogous structure of a graded algebra for outer derivations. A non-trivial graduation is obtained for all groups that are not perfect.