Affine actions on Nilpotent Lie groups

TL;DR

This paper investigates simply transitive affine actions of nilpotent Lie groups on other nilpotent groups, translating the problem to Lie algebra level and exploring special cases like abelian actions and low-dimensional scenarios.

Contribution

It introduces a Lie algebraic translation for the existence of simply transitive affine actions and characterizes abelian actions via LR-structures.

Findings

Translation of existence questions to Lie algebra level

Characterization of abelian affine actions through LR-structures

Analysis of low-dimensional cases (dim(G)=dim(N)<6)

Abstract

To any connected and simply connected nilpotent Lie group N, one can associate its group of affine transformations Aff(N). In this paper, we study simply transitive actions of a given nilpotent Lie group G on another nilpotent Lie group N, via such affine transformations. We succeed in translating the existence question of such a simply transitive affine action to a corresponding question on the Lie algebra level. As an example of the possible use of this translation, we then consider the case where dim(G)=dim(N) less than 6. Finally, we specialize to the case of abelian simply transitive affine actions on a given connected and simply connected nilpotent Lie group. It turns out that such a simply transitive abelian affine action on N corresponds to a particular Lie compatible bilinear product on the Lie algebra of N, which we call an LR-structure.