  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.
