Homogeneous symplectic 4-manifolds and finite dimensional lie algebras of symplectic vector fields on the symplectic 4-space

We classify the finite type (in the sense of E. Cartan theory of prolongations) subalgebras h ⊂ sp(V), where V is the symplectic 4-dimensional space, and show that they satisfy h(k) = 0 for all k > 0. Using this result, we reduce the problem of classification of graded transitive finite-dimensional Lie algebras g of symplectic vector fields on V to the description of graded transitive finite-dimensional subalgebras of the full prolongations p(∞)1 and p(∞)2, where p1 and p2 are the maximal parabolic subalgebras of sp(V). We then classify all such g ⊂ p(∞)i, i = 1, 2, under some assumptions, and describe the associated 4-dimensional homogeneous symplectic manifolds (M = G/K, Ω). We prove that any reductive homogeneous symplectic manifold (of any dimension) admits an invariant torsion free symplectic connection, i.e., it is a homogeneous Fedosov manifold, and give conditions for the uniqueness of the Fedosov structure. Finally, we show that any nilpotent symplectic Lie group (of any dimension) admits a natural invariant Fedosov structure which is Ricci-flat.