# Moscow Mathematical Journal

Volume 20, Issue 2, April–June 2020  pp. 217–256.

Homogeneous Symplectic 4-Manifolds and Finite Dimensional Lie Algebras of Symplectic Vector Fields on the Symplectic 4-Space

Authors:  D. V. Alekseevsky (1) and A. Santi (2)
Author institution:(1) A. A. Kharkevich Institute for Information Transmission Problems, B. Karetnyi per. 19, 127051, Moscow, Russia
University of Hradec Králové, Faculty of Science, Rokitanského 62, 50003 Hradec Králové, Czech Republic
(2) Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126, Bologna, Italy

Summary:

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 4-dimensional homogeneous symplectic 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 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.

2010 Math. Subj. Class. 53D05, 53C30, 17B66, 53C05.

Keywords: Homogeneous symplectic manifold, Lie algebra of symplectic vector fields, E. Cartan’s prolongation, homogeneous Fedosov manifold.