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

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 $\sll(2) \oplus \sll(m_1) \oplus \cdots \oplus \sll(m_k)$ 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 $\sll(2)$, $\sll(2) \oplus \sll(3)$, $\sll(2) \oplus \sll(3) \oplus \sll(11)$ for example.