Cohomology and deformations of the infinite dimensional filiform Lie algebra m_2

17 pagesInternational audienceLet m_2 denote the infinite dimensional N-graded Lie algebra defined by the basis e_i for i >= 1 and by relations [e_1,e_i]=e_{i+1} for all i >= 2, [e_2,e_j]=e_{j+2} for all j >= 3. We compute in this article the bracket structure on H^1(m_2;m_2), H^2(m_2;m_2) and in relation to this, we establish that there are only finitely many true deformations of m_2 in each weight by constructing them explicitely. It turns out that in weight 0 one gets only trivial and one formal non-converging deformations