A categorical approach to the study of derivations in group algebras

Abstract

We present a review of the results devoted to describing families of operators obeying some inductive identities (e. g. Leibniz’s rule — the case of derivations, Fox derivation, and (σ,τ)-derivations) as characters on a suitable groupoid. We first give an implementation of this construction for derivations in group algebras and Fox derivations as characters on an action groupoid. It is also demonstrated how this construction can be realized for derivations on algebras generated by Maltsev semigroups, for the case of derivations with values in finite rings, and for (σ,τ)-derivations.