Cohomology and deformations of the infinite dimensional filiform Lie algebra m_2

Details Cohomology and deformations of the infinite dimensional filiform Lie algebra m_2 Cohomology and deformations of the infinite dimensional filiform Lie algebra m_2

2007-08-02

arxiv.org/abs/0708.0363v2

Journal of Algebra 319 (2008), 5125-5143

Mathematics

Representation Theory

17 pages

Scientific paper

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.

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