Versal Deformations and Versality in Central Extensions of Jacobi's Schemes

Abstract

Let ${Ł}_m$ be the scheme of the laws defined by the Jacobi's identities on $\K^m$ with $\K$ a field. A deformation of $\g\inŁ_m$, parametrized by a local $\K$-algebra $\A$, is a local $\K$-algebra morphism from the local ring of ${Ł}_m$ at ${\varphi }_m$ to $\A$. The problem to classify 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 ${\varphi }_m=\mathrm{R}⋉{\varphi }_n$ in ${Ł}_m$ and its nilpotent radical ${\varphi }_n$ in the $\mathrm{R}$-invariant scheme ${Ł}_n^{\mathrm{R}}$ with reductive part $\mathrm{R}$, under some conditions. So the versal deformations of ${\varphi }_m$ in ${Ł}_m$ is deduced to those of ${\varphi }_n$ in ${Ł}_n^{\mathrm{R}}$, 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.

29 pages; Transformation Groups, 2009