Algorithme de construction des algèbres de Lie rigides

Abstract

An n -dimensional complex Lie algebra is rigid if its orbit under the canonical action of the full linear group is open in the variety defined by the Jacobi identities. The authors have perfected a method for obtaining solvable rigid Lie algebras independent of the classical cohomological criteria. This study is done in two steps: 1. First the authors reduce the field of research with conditions on these algebras (in particular on the root systems defined by the eigenvalues of the adjoint operator of a regular element). 2. Then they study the selected algebra locally with a perturbation in the framework of nonstandard analysis for establishing the rigidity. This method permits one to build many rigid laws; in particular it gives 18 rigid Lie algebras over C 10 having an operator with eigenvalues 1, 2, 3, 4, 5, 6, 7, 8, 0, 0. Note that the existence of a rich root system simplifies this study and reduces local calculations. Conversely, if the root system is null (nilpotent Lie algebra) then the study is entirely local, but we have not yet encountered such a rigid Lie algebra.