Low dimensional Lie groups admitting left invariant flat projective or affine structures

TL;DR

This paper classifies low-dimensional real Lie groups regarding their ability to admit flat projective or affine structures, establishing conditions based on dimension and algebraic properties.

Contribution

It proves that all real Lie groups of dimension five or less admit flat projective structures, and characterizes those with flat affine structures by their Lie algebra's perfection.

Findings

All real Lie groups of dimension ≤5 admit flat projective structures.

A Lie group admits a flat affine structure iff its Lie algebra is not perfect.

Provides a complete classification for low-dimensional cases.

Abstract

We prove that any real Lie group of dimension \leq 5 admits a left invariant flat projective structure. We also prove that a real Lie group L of dimension \leq 5 admits a left invariant flat affine structure if and only if the Lie algebra of L is not perfect.