Deformations of Filiform Lie Algebras and Symplectic Structures

D. V. Millionshchikov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology, discrete geometry, and set theory, Tome 252 (2006), pp. 194-216

Résumé

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.