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

TL;DR

This paper classifies certain finite-dimensional Lie subalgebras of symplectic vector fields on 4-space, leading to a comprehensive understanding of homogeneous symplectic 4-manifolds and their Fedosov structures.

Contribution

It provides a classification of finite type subalgebras of sp(4), reduces the classification of symplectic vector fields to maximal parabolic subalgebras, and establishes existence and uniqueness results for Fedosov structures on homogeneous symplectic manifolds.

Findings

Classified all finite type subalgebras of sp(4).

Reduced classification problem to subalgebras of maximal parabolic prolongations.

Proved existence of invariant torsion-free symplectic connections on homogeneous manifolds.

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.