Cohomology and deformations of the infinite dimensional filiform Lie algebra m_2

Abstract

Denote $\fm_2$ the infinite dimensional $\mathbb{N}$-graded Lie algebra defined by the basis $e_i$ for $i\ge 1$ and by relations $[e_1,e_i]=e_{i+1}$ for all $i\ge 2$, $[e_2,e_j]=e_{j+2}$ for all $j\ge 3$. We compute in this article the bracket structure on $H^1(\fm_2,\fm_2)$, $H^2(\fm_2,\fm_2)$ and in relation to this, we establish that there are only finitely many true deformations of $\fm_2$ in each weight by constructing them explicitely. It turns out that in weight 0 one gets as non-trivial deformation only one formal non-converging deformation.

17 pages