LR-algebras

TL;DR

This paper investigates the existence and structure of LR-algebras on Lie algebras, establishing their connection to 2-step solvability and classifying low-dimensional cases.

Contribution

It provides criteria for the existence of LR-structures, explores their properties, and classifies low-dimensional real LR-algebras.

Findings

Any Lie algebra with an LR-structure is 2-step solvable.

Certain classes of 2-step solvable Lie algebras admit LR-structures.

Classification of low-dimensional real LR-algebras.

Abstract

In the study of NIL-affine actions on nilpotent Lie groups we introduced so called LR-structures on Lie algebras. The aim of this paper is to consider the existence question of LR-structures, and to start a structure theory of LR-algebras. We show that any Lie algebra admitting an LR-structure is 2-step solvable. Conversely we find several classes of 2-step solvable Lie algebras admitting an LR-structure, but also classes not admitting such a structure. We study also ideals in LR-algebras, and classify low-dimensional real LR-algebras.