Cohomology and deformations of the infinite-dimensional filiform Lie algebra

Denote the infinite-dimensional -graded Lie algebra defined by the basis ei for i 1 and by relations [e1,ei]=ei+1 for all i 2, [e2,ej]=ej+2 for all j 3. We compute in this article the bracket structure on , and in relation to this, we establish that there are only finitely many true deformations of in each weight by constructing them explicitly. It turns out that in weight 0 one gets as non-trivial deformation only one formal non-converging deformation.