Versal deformations and versality in central extensions of Jacobi schemes

Abstract

Let \(\mathfrak{L}_m \) be the scheme of the laws defined by the Jacobi identities on \(\mathbb{K}^m \) with \(\mathbb{K}\) a field. A deformation of \(\mathfrak{g} \in \mathfrak{L}_m \), parametrized by a local ring A, is a local morphism from the local ring of \(\mathfrak{L}_m \) at ϕm to A. The problem of classifying all the deformation equivalence classes of a Lie algebra with given base is solved by “versal” deformations. First, we give an algorithm for computing versal deformations. Second, we prove there is a bijection between the deformation equivalence classes of an algebraic Lie algebra ϕm = R ⋉ φn in \(\mathfrak{L}_m \) and its nilpotent radical φn in the R-invariant scheme \(\mathfrak{L}_{n}^{\rm R} \) with reductive part R, under some conditions. So the versal deformations of ϕm in \(\mathfrak{L}_m \) are deduced from those of φn in \(\mathfrak{L}_{n}^{\rm R} \), which is a more simple problem. Third, we study versality in central extensions of Lie algebras. Finally, we calculate versal deformations of some Lie algebras.