Cohomology and deformations of the infinite-dimensional filiform Lie algebra m0

Denote m0 the infinite-dimensional N-graded Lie algebra defined by basis ei, i⩾1, and relations [e1,ei]=ei+1 for all i⩾2. We compute in this article the bracket structure on H1(m0,m0), H2(m0,m0) and in relation to this, we establish that there are only finitely many true deformations of m0 in each non-positive weight by constructing them explicitly. It turns out that in weight 0 one gets exactly the other two filiform Lie algebras.