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

Denote m(0) the infinite-dimensional N-graded Lie algebra defined by basis e(i), i >= 1, and relations [e(1),e(i)] = e(i+1) for all i >= 2. We compute in this article the bracket structure on H(1) (m(0), m(0)), H(2) (m(0) m(0)) and in relation to this, we establish that there are only finitely many true deformations of m(0) 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. (C) 2007 Elsevier Inc. All rights reserved.