Four dimensional symplectic Lie algebras

Abstract

In this paper we deal with symplectic Lie algebras. All symplectic structures are determined for dimension four and the corresponding Lie algebras are classified up to equivalence. Symplectic four dimensional Lie algebras are described either as solutions of the cotangent extension problem or as symplectic double extension of $\Bbb RR^2$ by $\Bbb RR$. The difference in the choice of a certain model lies on the existence or not of a lagrangian ideal. Moreover all extensions of a two dimensional Lie algebra are determined, and so all solutions (up to equivalence) of the cotangent extension problem are given in dimension two. By studying the adjoint representation we generalize to higher dimensions finding obstructions to the existence of symplectic forms. Finally, as an appendix we compute the authomorphisms of four dimensional symplectic Lie algebras and the cohomology over $\Bbb RR$ of the solvable real four dimensional Lie algebras.