2007/03/09 by Carles, Roger, Petit, Toukaiddine
#Algebraic Geometry (math.AG) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.math/0703277
Let Łm be the scheme of the laws defined by the identities of Jacobi on \Km. The local studies of an algebraic Lie algebra \g=R\ltimes\n in Łm and its nilpotent part \n in the scheme ŁnR of R-invariant Lie algebras on \Kn 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 Łm and ŁnR 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 Łm at \g to \A. So the study of deformations for a large class of Lie algebras \g in Łm is equivalent to that of \n in ŁnR "modulo" the actions of groups, which is a more simple problem. The laws of ŁnR are nilpotent with the choice of R and then we can construct these laws by central extensions. This corresponds to an induction on the schemes themselves ŁnR→Łn+1R. We restrict this study to a torus R=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 t2=0 (t≠ 0) and t5=0 (t4≠ 0).