Varieties of lie algebra laws

Abstract

This chapter reviews the results that are obtained recently in the study of Lie algebras from a geometrical point of view. Any Lie algebra law is considered as a point of an affine algebraic variety defined by the polynomial equations coming from the Jacobi identity for a given basis. This approach gives an explanation of the difficulties in classification problems concerning the classes of nilpotent and solvable Lie algebras and the relative facility of the classification of semi-simple Lie algebras. Isomorphic Lie algebras correspond to the laws belonging to the same orbit relative to the action of the general linear group and classification problems can be reduced to the classification of these orbits. The reason of the difficulties in the classification problems are discussed and precise the place and role of different classes of Lie algebras. The two classes of nilpotent Lie algebras, which are more or less models of nilpotent Lie algebras: those are filiform algebras and the characteristically nilpotent Lie algebras are focused. The study of varieties of Lie algebra laws is essentially based on the cohomological study of Lie algebras and on deformation theory.