Deformations of Lie algebras and Induction of Schemes

Abstract

Let $Ł_m$ be the scheme of the laws defined by the identities of Jacobi on $\K^m$. The local studies of an algebraic Lie algebra $\g=\mathrm{R}\ltimes\n$ in $Ł_m$ and its nilpotent part $\n$ in the scheme $Ł_n^{\mathrm{R}}$ of $\mathrm{R}$-invariant Lie algebras on $\K^n$ are linked. This comparison is made by means of slices, which are transversal subschemes to the orbits of $\g$ and $\n$ under the classical groups acting on $Ł_m$ and $Ł_n^{\mathrm{R}}$ respectively. We prove a reduction theorem saying that, under certain conditions on $\g$, the local rings of the slices at $\g$ and $\n$ are isomorphic. In particular, $\g$ is rigid if and only if is $\n$. In the formalism developed at beginning of this paper, a deformation of $\g$ with base a local ring $\A$ is a local morphism from the local ring of $Ł_m$ at $\g$ to $\A$. So the study of deformations for a large class of Lie algebras $\g$ in $Ł_m$ is equivalent to that of $\n$ in $Ł_n^{\mathrm{R}}$ "modulo" the actions of groups, which is a more simple problem. The laws of $Ł_n^{\mathrm{R}}$ are nilpotent with the choice of $\mathrm{R}$ and then we can construct these laws by central extensions. This corresponds to an induction on the schemes themselves $Ł_n^{\mathrm{R}}\toŁ_{n+1}^{\mathrm{R}}$. We restrict this study to a torus $\mathrm{R}=\mathrm{T}$ for certain slices. This leads to a concept of continuous families with the possibility to have nilpotent parameters $t$ (the schemes are generally not reduced). This gives an alternative formalism for the problem of obstructions classes in the theory of formal deformations of M.Gerstenhaber. Examples are given with $t^2=0$ ($t\neq 0$) and $t^5=0$ ($t^4\neq 0$).