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

Abstract

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