Cocycles in Lie Groups, Cochains and Regularity Problem

After the fundamental work of Livschitz in \cite{L1,L2}, various research directions emerged, among which the following stand out: (i) the study of cocycles with values in groups and semigroups beyond $\mathbb{R}$, as well as the investigation of corresponding regularity results;(ii) the analysis of how a certain degree of regularity ($C^{k}$ for $k=1,2,\ldots,\infty ,\omega$) of the cocycle can confer corresponding regularity to the solution of the cohomological equation; and (iii) the study of higher-dimensional cohomology naturally associated with the action of groups such as $\mathbb{Z}^{k}$ or $\mathbb{R}^{k}$.The aim of this article is to present, as self-contained as possible, a review of the natural generalizations of the notions of cocycles and cochains, as well as their corresponding results, in the study of cohomological equations.