Graded filiform Lie algebras and symplectic nilmanifolds

Abstract

We study symplectic (contact) structures on nilmanifolds that cor- respond to the filiform Lie algebras - nilpotent Lie algebras of the maximal length of the descending central sequence. We give a complete classification of filiform Lie algebras that possess a basis e1, . . . , en, (ei, ej) = cijei+j (N-graded Lie algebras). In particular we describe the spaces of symplectic cohomology classes for all even-dimensional algebras of the list. It is proved that a symplec- tic filiform Lie algebra g is a filtered deformation of some N-graded symplectic filiform Lie algebra g0. But this condition is not sufficient. A spectral sequence is constructed in order to answer the question whether a given deformation of a N-graded symplectic filiform Lie algebra g0 admits a symplectic structure or not. Other applications and examples are discussed.