Cohomologies of difference Lie groups and van Est theorem

TL;DR

This paper develops a cohomology theory for difference Lie groups, establishes a van Est theorem relating group and algebra cohomologies, and classifies abelian extensions using second cohomology.

Contribution

It introduces the notion of representations for difference Lie groups and connects them with difference Lie algebras, extending the classical theory.

Findings

Established a cohomology theory for difference Lie groups.

Proved a van Est theorem linking group and algebra cohomologies.

Classified abelian extensions via second cohomology group.

Abstract

A difference Lie group is a Lie group equipped with a difference operator, equivalently a crossed homomorphism with respect to the adjoint action. In this paper, first we introduce the notion of a representation of a difference Lie group, and establish the relation between representations of difference Lie groups and representations of difference Lie algebras via differentiation and integration. Then we introduce a cohomology theory for difference Lie groups and justify it via the van Est theorem. Finally, we classify abelian extensions of difference Lie groups using the second cohomology group as applications.