Cohomology and deformations of the infinite dimensional filiform Lie algebra m_2

TL;DR

This paper investigates the cohomology and deformation theory of an infinite-dimensional, graded Lie algebra called m_2, revealing the structure of its cohomology groups and identifying finitely many true deformations in each weight.

Contribution

It explicitly computes the cohomology groups of m_2 and establishes the finiteness of true deformations in each weight, including a specific non-converging deformation in weight zero.

Findings

Computed H^1 and H^2 cohomology groups of m_2

Established finiteness of true deformations per weight

Identified a unique non-converging deformation in weight zero

Abstract

Denote $\fm_2$ the infinite dimensional $\N$-graded Lie algebra defined by the basis $e_i$ for $i\geq 1$ and by relations $[e_1,e_i]=e_{i+1}$ for all $i\geq 2$, $[e_2,e_j]=e_{j+2}$ for all $j\geq 3$. 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.