ИСТИНА

We study symplectic structures on nilmanifolds (compact quotient spaces of nilpotent Lie groups over lattices) that correspond 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 the basis e1, ..., en, [ei,ej]=cijei+j (they are examples of N-graded Lie algebras). In particular we describe the space of symplectic cohomology classes for each algebra of the list. It is proved that a symplectic filiform Lie algebra g is a 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 an N-graded symplectic filiform Lie algebra g0 admit a symplectic structure or not. Other applications and examples are discussed.