Homogeneous spaces with invariant flat Cartan structures.

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

“…Let L be a real Lie group, and let l be its Lie algebra. Then Mendez, Lopera (Theorem 2.2 in [15]) showed that a left invariant flat real Cartan structure (P, ω) of type G/G ′ on L induces a real Lie algebra homomorphism f : l → g. The construction of homomorphism f is essentially taken from [1]. In the holomorphic category, we have the same assertion.…”

“…Proof. As we have seen above (P, ω) induces a Lie algebra homomorphism f = j * ω : l → g by Theorem 2.2 in [15]. Since the map j : L → P satisfies π • j = id, we have j * (T e L) ⊕ T ôπ −1 (ô) = T ôP .…”

“…In terms of Cartan structures, left invariance is described by the following: 15]). Let (P, ω) be a Cartan structure of type G/G ′ on L. Then (P, ω) is said to be left invariant if there exists a left action L × P ∋ (a, u) → L a ′ (u) ∈ P satisfying the following conditions: For any a ∈ L, L a ′ is a bundle isomorphism such that π P • L a ′ = L a • π P and L ′ a * ω = ω.…”

“…In this section we prove that left invariant flat Cartan structures of type G/G ′ on L correspond to certain injective Lie algebra homomorphisms of l to g, which we call simply transitive embeddings (cf. Theorem 2.3 in [15]). Recall that dim l = dim g/g ′ .…”