Cocycles in Lie Groups, Cochains and Regularity Problem

TL;DR

This paper reviews generalizations of cocycles and cochains in Lie groups, focusing on regularity results, higher-dimensional cohomology, and their implications for cohomological equations.

Contribution

It provides a comprehensive, self-contained overview of recent advances in cocycle theory, including new perspectives on regularity and higher cohomology in Lie groups.

Findings

Analysis of cocycles with values beyond R

Regularity transfer from cocycles to solutions

Extension of cohomology concepts to higher dimensions

Abstract

After the fundamental work of Livschitz in [1; 2], various research directions emerged, among which the following stand out: (i) the study of cocycles with values in groups and semigroups beyond R, as well as the investigation of corresponding regularity results; (ii) the analysis of how a certain degree of regularity 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 other than Z or R. 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.