Prehomogeneous Affine Representations and Flat Pseudo-Riemannian Manifolds

TL;DR

This paper explores the relationship between prehomogeneous affine representations of Lie groups and flat Pseudo-Riemannian manifolds, developing characteristic class theory and providing applications to various flat affine structures.

Contribution

It introduces a comprehensive theory linking prehomogeneous affine representations with flat pseudo-Riemannian and symplectic manifolds, including characteristic classes and applications.

Findings

Development of characteristic classes for prehomogeneous affine representations

Applications to flat affine and Pseudo-Riemannian manifolds

Insights into symplectic affine flat manifolds

Abstract

The theory of flat Pseudo-Riemannian manifolds and flat affine manifolds is closely connected to the topic of prehomogeneous affine representations of Lie groups. In this article, we exhibit several aspects of this correspondence. At the heart of our presentation is a development of the theory of characteristic classes and characters of prehomogeneous affine representations. We give applications concerning flat affine, as well as Pseudo-Riemannian and symplectic affine flat manifolds.