Flat Lie groups, Frobenius Lie algebras and \'{e}tale prehomogeneous vector spaces for reductive Lie groups

TL;DR

This paper explores the deep connections between flat connections on Lie groups, algebraic structures like Frobenius Lie algebras, and prehomogeneous vector spaces, providing classifications and establishing correspondences for reductive Lie groups.

Contribution

It establishes a one-to-one correspondence between left-symmetric Lie algebras with a right identity and étale prehomogeneous vector spaces, and classifies these spaces for groups with simple Levi factors.

Findings

Any left-symmetric structure on a reductive Lie algebra has a right identity.

Classification of flat connections on reductive Lie groups reduces to classifying étale prehomogeneous vector spaces.

Complete classification of étale prehomogeneous vector spaces for groups with simple Levi factors.

Abstract

In this paper, we established the relationship among left-invariant flat connections on Lie groups, left-symmetric algebras, Frobenius Lie algebras and \'{e}tale prehomogeneous vector spaces, gave a one-to-one correspondence between the left-symmetric Lie algebras with a right identity and the \'{e}tale prehomogeneous vector spaces for a Lie group, and proved that, in essence, any left-symmetric structure on a reductive Lie algebra has a right identity, which implies that the classification of flat connections on a reductive Lie group $G$ amounts to that of \'{e}tale prehomogeneous vector spaces for $G$. We classified the \'{e}tale prehomogeneous vector spaces for $G$ with simple Levi factors.