A new look at Lie algebras

Abstract

We present a new look at the description of real finite-dimensional Lie algebras. The basic ingredient is a pair (F,v) consisting of a linear mapping F∈End(V) with an eigenvector v. This pair allows to build a Lie bracket on a dual space to a linear space V. The Lie algebra obtained in this way is solvable. In particular, when F is nilpotent, the Lie algebra is actually nilpotent. We show that these solvable algebras are the basic bricks of the construction of all other Lie algebras. Using relations between the Lie algebra, the Lie–Poisson structure and the Nambu bracket, we show that the algebra invariants (Casimir functions) are solutions of an equation which has an interesting geometric significance. Several examples illustrate the importance of these constructions.