Cohomology of n-categories and derivations in group algebras

TL;DR

This paper develops a cohomological framework for n-groupoids and n-characters, providing new insights into derivations in group algebras and constructing examples through a novel approach.

Contribution

It introduces a new cohomology-based method for studying derivations in group algebras using n-groupoids and n-characters, expanding theoretical understanding.

Findings

Constructed a sequence of spaces of n-characters and morphisms

Applied the framework to describe external derivations in group algebras

Provided new examples of derivations using the developed approach

Abstract

This work represents the concept of an n-groupoid $ \Gamma^n $ and n-characters $ \chi_n $ on n-groupoids as complex-valued maps from spaces of different classes of morphisms satisfying the condition $ \chi_n (\psi \circ_k \varphi) = \chi_n (\psi) + \chi_n (\varphi) $ for any possible compositions. A sequence of spaces of n-characters and morphisms between them is constructed and its accuracy is shown. This construction has important application for describing the derivations in a group algebras. In particular, this approach allows us to study the algebra of external derivations from a new point of view, and also to construct some interesting examples. The work was carried out under the guidance of Arutyunov A. A.. And it is based on the Mishchenko A. S. ideas.