Complete LR-structures on solvable Lie algebras

TL;DR

This paper characterizes solvable Lie algebras that admit complete LR-structures, showing that the existence of any LR-structure implies the existence of a complete one, thus clarifying their structural properties.

Contribution

It proves that for solvable Lie algebras, admitting a complete LR-structure is equivalent to admitting any LR-structure, providing a complete characterization.

Findings

Complete LR-structures exist if and only if any LR-structure exists.

The result applies specifically to solvable Lie algebras.

Clarifies the relationship between LR-structures and Lie algebra properties.

Abstract

An LR-structure on a Lie algebra is a bilinear product, satisfying certain commutativity relations, and which is compatible with the Lie product. LR-structures arise in the study of simply transitive affine actions on Lie groups. In particular one is interested in the question which Lie algebras admit a complete LR-structure. In this paper we show that a Lie algebra admits a complete LR-structure if and only if it admits any LR-structure.