Derivations in Group Algebras

Derivations on a group algebra (i.e., linear operators satisfying the Leibniz rule) can be described as characters on the groupoid of the adjoint action. At the same time, characters (i.e., maps on morphisms such that for a pair of composable morphisms the condition $\chi(\psi\circ \varphi) = \chi(\psi)+\chi(\varphi)$ holds) can be described using special graphs analogous to Cayley graphs: here, the left action by multiplication is replaced by conjugation. The talk will be structured around these two constructions. Among other things, we will consider the following points in detail.
In the case of a commutative group, we will give a complete description of all possible derivations (they will coincide with the so-called central derivations) using characters of the group (homomorphisms into the additive group of complex numbers). For non-commutative finite groups and FC-groups (i.e., groups where all conjugacy classes are finite), we will show that all derivations are inner (this is a well-known fact, however we will give a simple categorical proof). In the general case, we will show that the ideal of inner derivations can be extended quite naturally to the so-called quasi-inner derivations (they can be understood as a formal closure of inner derivations), and their structure turns out to be closely related to coarse invariants, such as the number of ends.
In conclusion, we will give a brief overview of known generalizations of this construction: replacing the Leibniz rule with weaker conditions, replacing the coefficients of the group algebra with various rings, including those of finite characteristic.