The groupoid description of derivations in group algebras

The paper is based on the approach proposed in [1] jointly with A.S. Mishchenko and A.I. Stern. The idea is that every derivation can be compared to some character on the groupoid of an adjoint action. By character we mean mappings $\chi\colon\mathrm{Hom}(\Gamma)\to\mathbb C$ such that $\chi(\psi\circ\phi)= \chi(\psi)+\chi(\phi)$ for any pair of connected morphisms. So the problem of the study of derivations is reduced to the study of characters on a groupoid (satisfying some additional property of local finiteness, but these are details). In this case it turns out that such a study is already reduced to ordinary group theory questions. In this case it turns out to be useful to consider not only the usual inner (and factor by them – outer) derivations, but also the so-called quasi-inner derivations – which can be understood as "infinite formal sums of inner". The very fact that a formally divergent sum of inner derivations can give an operator still valid in group algebra and which is a derivation (but it is not inner derivation) is quite interesting. Moreover, quasi-inner derivations form an ideal, so it is possible to decompose the algebra of differentiations into quasi-internal and quasi-external differentiations.
On the other hand, it turns out that if we take other variants of the groupoid of action (groups on themselves) as a groupoid, it is possible to construct other variants of operators satisfying some identities similar to Leibniz's rule. In particular, I will demonstrate a way to construct in this way another variant of the differential calculus of the group (the so-called Fox derivatives). In this way different theories can be combined into a single construction: the main thing is to choose the right groupoid.
The construction as a whole turns out to be quite general. It is possible to study in this way also derivations in some other types of associative algebras as well as more general variants of derivations. I will also try to formulate some overview of such results in the talk.
[1] A. A. Arutyunov, A. S. Mishchenko, A. I. Shtern, “Derivations of group algebras”, Fundam. Prikl. Mat., 21:6 (2016), 65–78