I am studying nilpotent Lie algebra theory. The subject is really new to me and I am studying by myself(I have not a good advisor). I'd love your help with this.
Let $\mathfrak{n}$ be a finite-dimensional nilpotent Lie algebra (over an algebraically closed field of characteristic zero) and let $\operatorname{Der}(\mathfrak{n})$ be the algebra of derivations of $\mathfrak{n}$. The system of weights of $\mathfrak{n}$ is defined as being that of the natural representation of a "maximal torus" $T$ in $\operatorname{Der}(\mathfrak{n})$ and the $\operatorname{rank}$ is the dimension of $T$. By remarkable result due to Gabriel Favre (see [F]), it is known that for a fixed integer $n$ there are finitely systems of weights. Let $T$ be a system of weights, we denote by $\mathrm{N}(T)$ the class of those Lie algebras having the system of weights $T$.
My questions are:
- For a fixed integer $n$, are these system of weights classified?
- For a fixed integer $n$, can rank-one system of weights explicitly written?
- Are classified rank-one system of weights $T$ such that $\sharp\mathrm{N}(T)=1$
- Is there a good book or resource for learning about this topic and in general, about nilpotent Lie algebras (over $\mathbb{C}$ or $\mathbb{R}$)?
Any help is much appreciated!
[F] Favre, G.: Système de poids sur une algèbre de Lie nilpotente. Manuscripta Math. 9 (1973), 53-90.
