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

Abstract

We classify the finite type (in the sense of E. Cartan theory of prolongations) subalgebras $\mathfrak{h}\subset \mathfrak{sp}(V)$, where $V$ is the symplectic 4-dimensional space, and show that they satisfy ${\mathfrak{h}}^{(k)}=0$ for all $k>0$. Using this result, we reduce the problem of classification of graded transitive finite-dimensional Lie algebras $\mathfrak{g}$ of symplectic vector fields on $V$ to the description of graded transitive finite-dimensional subalgebras of the full prolongations ${\mathfrak{p}}_1^{(\infty )}$ and ${\mathfrak{p}}_2^{(\infty )}$, where ${\mathfrak{p}}_1$ and ${\mathfrak{p}}_2$ are the maximal parabolic subalgebras of $\mathfrak{sp}(V)$. We then classify all such $\mathfrak{g}\subset {\mathfrak{p}}_i^{(\infty )}$, $i=1,2$, under some assumptions and describe the associated homogeneous symplectic 4-manifolds $(M=G/K,\omega )$. 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 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.

37 pages, 2 tables; v4: minor comments, few typos corrected and one reference added. To appear on Mosc. Math. J

References in corpus (1)

• Symplectic and contact Lie algebras with an application to Monge-Ampere equations