Flat Affine or Projective Geometries on Lie Groups

TL;DR

This paper investigates which Lie groups admit flat affine or projective structures, especially focusing on bi-invariant cases, and explores their role in homogeneous spaces, partially answering a question about affine transformation groups.

Contribution

It characterizes Lie groups with flat bi-invariant affine or projective structures and examines their significance in homogeneous space theory.

Findings

Identifies Lie groups with flat bi-invariant affine or projective structures.

Provides partial positive answer to Medina's question on affine transformation groups.

Highlights the role of these groups in invariant structures on homogeneous spaces.

Abstract

This paper deals essentially with affine or projective transformations of Lie groups endowed with a flat left invariant affine or projective structure. These groups are called flat affine or flat projective Lie groups. Our main results determine Lie groups admitting flat bi-invariant affine or projective structures. These groups could play an essential role in the study of homogeneous spaces $M=G/H$ admitting flat affine or flat projective structures invariant under the natural action of $G$ on $M$. A. Medina asked several years ago if the group of affine transformations of a flat affine Lie group is a flat projective Lie group. In this work we provide a partial possitive answer to this question.