Locally Homogeneous Manifolds Defined by Lie Algebra of Infinitesimal Affine Transformations

This article deals with Lie algebra G of all infinitesimal affine transformations of the manifold M with an affine connection, its stationary subalgebra ℌ ⊂ G , the Lie group G corresponding to the algebra G , and its subgroup H ⊂ G corresponding to the subalgebra ℌ ⊂ G . We consider the center ℨ ⊂ G and the commutant [ G , G ] of algebra G . The following condition for the closedness of the subgroup H in the group G is proved. If ℌ ∩ ℨ + G ; G = ℌ ∩ [ G ; G ], then H is closed in G . To prove it, an arbitrary group G is considered as a group of transformations of the set of left cosets G / H , where H is an arbitrary subgroup that does not contain normal subgroups of the group G . Among these transformations, we consider right multiplications. The group of right multiplications coincides with the center of the group G . However, it can contain the right multiplication by element 𝒽 ¯ , belonging to normalizator of subgroup H and not belonging to the center of a group G . In the case when G is in the Lie group, corresponding to the algebra G of all infinitesimal affine transformations of the affine space M and its subgroup H corresponding to its stationary subalgebra ℌ ⊂ G , we prove that such element 𝒽 ¯ exists if subgroup H is not closed in G . Moreover 𝒽 ¯ belongs to the closures H ¯ of subgroup H in G and does not belong to commutant G , G of group G . It is also proved that H is closed in G if P + ℨ ∩ ℌ = P ∩ ℌ for any semisimple algebra P ∈ G for which P + ℜ = G .