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We show the correspondence between left invariant flat projective structureson Lie groups and certain prehomogeneous vector spaces. Moreover by using theclassification theory of prehomogeneous vector spaces, we classify complex Liegroups 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 projectivestructures 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) \oplussl(11) for example.

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