On derivations in group algebras and other algebraic structures

Abstract

The work is devoted to a survey of known results related to the study of derivationsin group algebras, bimodules and other algebraic structures, as well as to various generalizationsand variations of this problem. A review of results on derivations in L_1 (G) algebras, invon Neumann algebras, and in Banach bimodules is given. Algebraic problems are discussed,in particular, derivations in groups, (σ,τ)-derivations, and the Fox calculus. A review ofsome results related to the application to pseudodifferential operators and the constructionof the symbolic calculus is also given. In conclusion, some results related to the description ofderivations as characters on the groupoid of the adjoint action are described. Some applicationsare also described: to coding theory, the theory of ends of metric spaces, and rough geometry.