On the rigidity of solvable Lie algebras

Abstract

The author gives a method for the construction of solvable rigid Lie algebras in the varieties defined by Jacobi's identities. His method does not use tables of nilpotent Lie algebras and the rigidity is not obtained by usual cohomological criteria. He uses the fact that such an algebra g 0 is algebraic and has a Chevalley splitting T⊕n , where n is the maximal nilpotent ideal and T is a subalgebra constituted by ad g 0 -semisimple elements (by the adjoint representation ad g 0 we can also identify T to a maximal torus of derivations over n) . The author takes a regular semisimple element X (X∈T) and studies all possible systems of relations (S) checked for the eigenvalues of the operator ad g 0 X . Certain systems (S) are not suitable and are directly eliminated. For the remaining cases, the rigidity is established by the method of perturbations in nonstandard analysis. As applications of his method the author gives a classification of solvable rigid Lie algebras of dimension ≤8 and a list of 48 rigid algebras in dimension 9.