Deformations of Filiform Lie Algebras and Symplectic Structures

We study symplectic structures on filiform Lie algebras, which are nilpotent Lie algebras with the maximal length of the descending central sequence. Let $\mathfrak g$ be a symplectic filiform Lie algebra and $\dim \mathfrak g=2k\ge 12$. Then $\mathfrak g$ is isomorphic to some $\mathbb N$-filtered deformation either of $\mathfrak m_0(2k)$ (defined by the structure relations $[e_1,e_i]=e_{i+1}$, $i=2,\dots ,2k-1$) or of $\mathcal V_{2k}$, the quotient of the positive part of the Witt algebra $W_+$ by the ideal of elements of degree greater than $2k$. We classify $\mathbb N$-filtered deformations of $\mathcal V_n$: $[e_i,e_j]=(j-i)e_{i+1}+\sum _{l\ge 1}c_{ij}^l e_{i+j+l}$. For $\dim \mathfrak g=n \ge 16$, the moduli space $\mathcal M_n$ of these deformations is the weighted projective space $\mathbb K\mathrm P^4(n-11,n-10,n-9,n-8,n-7)$. For even $n$, the subspace of symplectic Lie algebras is determined by a single linear equation.