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

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.