Déformations et éléments nilpotents dans les schémas définis par les identités de Jacobi. (Deformations and nilpotent elements in the schemes defined by Jacobi’s identities.). (French) Zbl 0734.17008

From the author’s abstract: A deformation of a Lie algebra law \(\phi\), parametrized by a local ring A, is a morphism \({\mathcal O}\to A\), where \({\mathcal O}\) is the local ring at \(\phi\) in the scheme defined by Jacobi’s identities. When A is complete, this is equivalent to a deformation in which the structure constants inducing local parameters on the orbit of \(\phi\) (under the canonical action of the full linear group) are fixed. This applies to the universal deformation defined by the identity. Thus, we obtain deformations where only appear parameters expressing the variation of the orbit, as well as nilpotent elements in the completion of \({\mathcal O}\). The nonzero obstruction in M. Gerstenhaber’s theory of formal deformations corresponds here to the nilpotency of a parameter. We give examples of such deformations, with nilpotent elements of order 2 and 5.
Reviewer: F.Rouvière (Nice)


17B56 Cohomology of Lie (super)algebras
14D15 Formal methods and deformations in algebraic geometry