Deformations of crossed homomorphisms on Lie groups

TL;DR

This paper investigates how crossed homomorphisms on Lie groups deform, using cohomology to establish rigidity results and exploring their relationship with Lie algebra cohomology, especially in low-dimensional cases.

Contribution

It introduces a cohomological framework for analyzing deformations of crossed homomorphisms on Lie groups and characterizes their rigidity properties in specific cases.

Findings

Rigidity results for crossed homomorphisms on Lie groups

Relationship between cohomology of crossed homomorphisms on Lie groups and Lie algebras

Characterization of rigidity on two-dimensional Lie groups

Abstract

In this paper, we study deformations of crossed homomorphisms on Lie groups by means of the cohomology which controls them. Using the Moser type argument, we obtain several rigidity results of crossed homomorphisms on Lie groups. We further investigate the relationship between the cohomology of crossed homomorphisms on Lie groups and that on Lie algebras. Finally, we characterize the rigidity properties of all crossed homomorphisms with respect to the conjugation action on the connected and simply connected two-dimensional Lie group.