Derivations of Group Algebras

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

Derivations of group rings for finite and FC groups

“…In this paper we apply the the character technique approach, which is proposed in [3], to the case when group has finite conjugacy classes. As a consequence, the character technique gives tools to prove a decomposition theorem for Der(A[G]) in case G is an FC-group and A is a commutative unital ring.…”

On $(σ,τ)$-derivations of group algebra as category characters

“…Here we apply an approach developed in [4,6,7,44] to describe derivation in terms of groupoid geometrical properties. We construct a groupoid Γ associated with the group algebra in the following way:…”

“…It is important to note that in the cited publications derivations are considered in topological context of the classes of algebras and modules equipped with normed or more general topological structures. If we consider the problem in algebraic way it is easy to find examples of non-inner derivations [4][5][6]. Algebraic view on the derivation problem is also presented in [3] together with more complete bibliography.…”

“…In this paper, we consider (σ , τ)-derivations and σ -derivations of the group algebra C[G] of a discrete countable group G. We apply the approach proposed in [4][5][6][7]44] for the description of derivations in group algebras to study the space of (σ , τ)-derivations of the group algebras.…”

Derivations of Group Algebras and Hochschild cohomology

“…This talk is motivated by comparing the results of our last work, Arutyunov A.A., Mishchenko A.S.,(2018), [3], (the full text is presented in the journal "Matematicheskij sbornik" [4]) and also works by Arutyunov, A.A., Mishchenko, A.S., Shtern, A.I., (2016), [5], [6], in which a description of the algebra of exterior derivations of the group algebra R[G] of a finitely representable discrete group G is presented in terms of the Cayley complex of the groupoid G of the adjoint action of the group G , with the results of Burghelea (1985) [7] and Benson (1995Benson ( , 1991 [1], [2], which describe the Hochschild homology and cohomology of the group algebra R[G] in terms of the classifying spaces BC x centralizers C x of conjugate classes x of groups G .…”