Left invariant flat projective structures on Lie groups and prehomogeneous vector spaces

TL;DR

This paper establishes a link between flat projective structures on Lie groups and prehomogeneous vector spaces, classifies complex Lie groups with such structures, and provides explicit conditions for their existence.

Contribution

It introduces a correspondence between flat projective structures and prehomogeneous vector spaces and classifies complex Lie groups with irreducible structures using this framework.

Findings

Certain Lie groups admit flat projective structures based on a specific algebraic condition.

Explicit classification of complex Lie groups with these structures.

Examples include sl(2), sl(2)  sl(3), sl(2)  sl(3)  sl(11).

Abstract

We show the correspondence between left invariant flat projective structures on Lie groups and certain prehomogeneous vector spaces. Moreover by using the classification theory of prehomogeneous vector spaces, we classify complex Lie groups admitting irreducible left invariant flat complex projective structures. As a result, direct sums of special linear Lie algebras sl(2) \oplus sl(m_1) \oplus \cdots \oplus sl(m_k) admit left invariant flat complex projective structures if the equality 4 + m_1^2 + \cdots + m_k^2 -k - 4 m_1 m_2 \cdots m_k = 0 holds. These contain sl(2), sl(2) \oplus sl(3)$, sl(2) \oplus sl(3) \oplus sl(11) for example.