Système de poids sur une algèbre de Lie nilpotente

Let g be a nilpotent Lie algebra (of finite dimension n over an algebraically closed field of characteristic zero) and let Der(g) be the algebra of derivations of g. The system of weights of g is defined as being that of the standard representation of a "maximal torus" in Der(g). For a fixed integer n, it is well-known that there are in general uncountably many isomorphism classes of nilpotent Lie algebra of dimension n; but we show that there are finitely many systems of weights, and each of them is explicitely constructed. The class of those Lie algebras having a given (arbitrary) system of weights is also studied. The first chapter is a setting for the study of nilpotent Lie algebras, used to prove some general theorems. In the second chapter, attention is restricted to a class of nilpotent Lie algebras for which our setting is particularly well adapted.


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