Cohomology and deformations of the infinite dimensional filiform Lie algebra m_0

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 H1(m_0,m_0), H2(m_0,m_0) and in relation to this, we establish that there are only finitely many true deformations of m_0 in each nonpositive weight, by constructing them explicitely. It turns out that in weight 0 one gets exactly the other two filiform Lie algebras.Comment: 25 page

Construction of Miniversal Deformations of Lie Algebras

We consider deformations of finite or infinite dimensional Lie algebras over a field of characteristic 0. There is substantial confusion in the literature if one tries to describe all the non-equivalent deformations of a given Lie algebra. It is known that there is in general no "universal" deformation of the Lie algebra L with a commutative algebra base A with the property that for any other deformation of L with base B there exists a unique homomorphism f: A -> B that induces an equivalent deformation. Thus one is led to seek a "miniversal" deformation. For a miniversal deformation such a homomorphism exists, but is unique only at the first level. If we consider deformations with base spec A, where A is a local algebra, then under some minor restrictions there exists a miniversal element. In this paper we give a construction of a miniversal deformation.Comment: 29 pages, (plain) Te