Deformations of filiform Lie algebras and symplectic structures

Abstract

We study symplectic structures on filiform Lie algebras, which are niplotent Lie algebras with the maximal length of the descending central sequence. Let g be a symplectic filiform Lie algebra and dim g = 2k ≥ 12. Then g is isomorphic to some ℕ-filtered deformation either of m0(2k) (defined by the structure relations [e1, e i ] = ei+1, i = 2,…, 2k − 1) or of V2k, the quotient of the positive part of the Witt algebra W+ by the ideal of elements of degree greater than 2k. We classify ℕ-filtered deformations of V n : [e i , e j ] = (j − i)ei+1 + Σl≥1c ij l e i+j+l . For dim g = n ≥ 16, the moduli space ℳn of these deformations is the weighted projective space \(\mathbb{K}P^4 \left( {n - 11,n - 10,n - 9,n - 8,n - 7} \right)\). For even n, the subspace of symplectic Lie algebras is determined by a single linear equation.