On $(\sigma,\tau)$-derivations of group algebra as category characters

TL;DR

This paper generalizes the decomposition theorem for $(\sigma, au)$-derivations of group algebras, using groupoids and characters, and explores conditions for all such derivations to be inner, especially in specific group classes.

Contribution

It introduces a generalized decomposition theorem for $(\sigma, au)$-derivations on group algebras using algebraic and categorical tools, extending prior results on ordinary derivations.

Findings

Decomposition theorem for $(\sigma, au)$-derivations established

Conditions identified for all derivations to be inner

Detailed analysis for $(\sigma, au)$-nilpotent and $(\sigma, au)$-$FC$ groups

Abstract

For the space of $(\sigma,\tau)$-derivations of the group algebra $ \mathbb{C} [G] $ of discrete countable group $G$, the decomposition theorem for the space of $(\sigma,\tau)$-derivations, generalising the corresponding theorem on ordinary derivations on group algebras, is established in an algebraic context using groupoids and characters. Several corollaries and examples describing when all $(\sigma,\tau)$-derivations are inner are obtained. Considered in details cases on $(\sigma,\tau)-$nilpotent groups and $(\sigma,\tau)$-$FC$ groups.