Roger Carles | Searcharxiv

arXiv2010

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

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 $ϕ_m$ 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 $ϕ_m=\mathrm{R}\ltimesϕ_n$ in $Ł_m$ and its nilpotent radical $ϕ_n$ in the $\mathrm{R}$-invariant scheme $Ł_n^{\mathrm{R}}$ with reductive part $\mathrm{R}$, under some conditions. So the versal deformations of $ϕ_m$ in $Ł_m$ is deduced to those of $ϕ_n$ 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.

math.RA↗

arXiv2007

Deformations of Lie algebras and Induction of Schemes

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 $t^5=0$ ($t^4\neq 0$).

math.AG↗