Cohomology and deformations of the infinite-dimensional filiform Lie algebra m2

Denote m2 the infinite-dimensional N-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 H1(m2,m2), H2(m2,m2) and in relation to this, we establish that there are only finitely many true deformations of m2 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.