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) ⊕ sl(m1) ⊕ · · · ⊕ sl(m k ) admit left invariant flat complex projective structures if the equality 4 + m 2 1 + · · · + m 2 k − k − 4m1m2 · · · m k = 0 holds. These contain sl(2), sl(2) ⊕ sl(3), sl(2) ⊕ sl(3) ⊕ sl(11) for example.2000 Mathematics Subject Classification. 53A20, 11S90. Key words and phrases. left invariant flat projective structure; prehomogeneous vector space.

We prove that any real Lie group of dimension ≤ 5 admits a left invariant flat projective structure. We also prove that a real Lie group L of dimension ≤ 5 admits a left invariant flat affine structure if and only if the Lie algebra of L is not perfect.

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A special linear Lie group over the real number field and the quarternion field admits a projectivley flat affine connection. We show that parabolic subgroups are autoparallel submanifolds and give a criterion the induced connection is projectively equivalent to a flat affine connection.

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