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I am studying nilpotent Lie algebra theory. The subject is really new to me and I am studying by myself. 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:

  1. For a fixed integer $n$, are these system of weights classified?
  2. For a fixed integer $n$, can rank-one system of weights explicitly written?
  3. Are classified rank-one system of weights $T$ such that $\sharp\mathrm{N}(T)=1$
  4. 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.

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3 Answers

1.) No, the weight systems are not classified in general. However, the weight systems for complex nilpotent Lie algebras of dimension $n\le 7$ have been computed by Roger Carles, in "Weight systems for complex nilpotent Lie algebras and application to the varieties of Lie algebras", in $1996$. Another reference, in addition to Pasha's references, is this paper of Magnin of $1998$.

2.), 3.) I think no.

4.) There are several books (e.g., by Goze, Onischik, Vinberg, etc.). Also, several articles discuss nilpotent Lie algebras in general, e.g. the article by E. M. Luks, What is a typical nilpotent Lie algebra ? For faithful representations of nilpotent Lie algebras, see, for example, here.

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Merely some further references, doesn't fit into the comment field:

G.F. Leger and E.M. Luks, Cohomology and weight systems for nilpotent Lie algebras, Bull. Amer. Math. Soc. 80 (1974), 77-80 http://projecteuclid.org/euclid.bams/1183535294

L.J. Santharoubane, Kac-Moody Lie algebras and the classification of nilpotent Lie algebras of maximal rank, Canad. J. Math. 34 (1982), 1215-1239 DOI:10.4153/CJM-1982-084-5

L.J. Santharoubane, Kac-Moody Lie algebras and the universal element for the category of nilpotent Lie algebras, Math. Ann. 263 (1983), 365-370 http://resolver.sub.uni-goettingen.de/purl?GDZPPN002323583

L. Magnin, Remarks on weight systems on cohomology of nilpotent Lie algebras, Algebras Groups Geom. 9 (1992), No.2, 111-135

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I don't know anything about weights in the nilpotent case, so this constitutes an answer to question 4. Classifying nilpotent Lie algebras is, as far as I know, a pretty "wild" problem. Nonetheless, there are some qualitative things which can be said. In particular, the category of nilpotent Lie algebras over a field $k$ of characteristic zero is equivalent to the category of unipotent groups over $k$, thanks to the Campbell-Hausdorff series, which expresses the exponential of the Lie bracket as a "formal Lie power series." Because we are dealing with nilpotent Lie algebras this power series has some finite degree, so there are no convergence issues.

One interesting consequence of this for representation theory is that when $k = \mathbb{Q}_p$ or $\mathbb{R}$, $\mathfrak{g}$ is a nilpotent Lie algebra over $k$, and $G$ is the unipotent group corresponding to $\mathfrak{g}$ endowed with the topology from $k$, we get a concrete geometrical description of the so-called dual space $\hat{G}$ of $G$. By definition $\hat{G}$ is the space of irreducible "nice" representations of $G$, where "nice" could mean unitary in either case or smooth when $k = \mathbb{Q}_p$, given the natural Fell topology (which has an unpleasantly complicated description). Then there is a canonical homeomorphism of $\hat{G}$ with the space of orbits of $G$ in $\mathfrak{g}^* = \text{Hom}_k(\mathfrak{g},k)$ under the so-called coadjoint action. For more on this, see Kirillov's notes here.

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