ИСТИНА

We study left-invariant symplectic and affine structures on nilpotent Lie groups that correspond to filiform Lie algebras – nilpotent Lie algebras of the maximal length of the descending central sequence. Symplectic filiform Lie algebras in large dimensions can be described as special deformations of two series of graded filiform Lie algebras. We study deformations of one of them: ’finite positive part’ of Virasoro algebra, i.e. a graded Lie algebra with basis e1, . . . , en structure relations of the following form: [ei, ej ] = (j−i)ei+j , i + j  n. For dimensions n  16 the moduli spaceMn of these deformations can be identified with the weighted projective space KP4(n−11, n−10, n−9, n−8, n−7) and for even dimensions n the subspace of symplectic Lie algebras is determined by one linear equation. The relation to the problem of existence of affine structure is discussed.