System of weights for nilpotent Lie algebras - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T03:00:15Z http://mathoverflow.net/feeds/question/70569 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70569/system-of-weights-for-nilpotent-lie-algebras System of weights for nilpotent Lie algebras Jennifer López Rodríguez 2011-07-17T21:39:44Z 2011-10-04T02:36:43Z <p>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.</p> <p>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 $T$.</p> <p>My questions are:</p> <ol> <li>For a fixed integer $n$, are these system of weights classified?</li> <li>For a fixed integer $n$, can rank-one system of weights explicitly written?</li> <li>Are classified rank-one system of weights $T$ such that $\sharp\mathrm{N}(T)=1$</li> <li>Is there a good book or resource for learning about this topic and in general, about nilpotent Lie algebras (over $\mathbb{C}$ or $\mathbb{R}$)?</li> </ol> <p>Any help is much appreciated!</p> <p>[F] Favre, G.: Système de poids sur une algèbre de Lie nilpotente. Manuscripta Math. 9 (1973), 53-90.</p> http://mathoverflow.net/questions/70569/system-of-weights-for-nilpotent-lie-algebras/72396#72396 Answer by Pasha Zusmanovich for System of weights for nilpotent Lie algebras Pasha Zusmanovich 2011-08-08T20:51:22Z 2011-08-08T22:16:35Z <p>Merely some further references, doesn't fit into the comment field: </p> <p>G.F. Leger and E.M. Luks, Cohomology and weight systems for nilpotent Lie algebras, Bull. Amer. Math. Soc. 80 (1974), 77-80 <a href="http://projecteuclid.org/euclid.bams/1183535294" rel="nofollow">http://projecteuclid.org/euclid.bams/1183535294</a></p> <p>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</p> <p>L.J. Santharoubane, Kac-Moody Lie algebras and the universal element for the category of nilpotent Lie algebras, Math. Ann. 263 (1983), 365-370 <a href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN002323583" rel="nofollow">http://resolver.sub.uni-goettingen.de/purl?GDZPPN002323583</a></p> <p>L. Magnin, Remarks on weight systems on cohomology of nilpotent Lie algebras, Algebras Groups Geom. 9 (1992), No.2, 111-135</p> http://mathoverflow.net/questions/70569/system-of-weights-for-nilpotent-lie-algebras/77096#77096 Answer by Justin Campbell for System of weights for nilpotent Lie algebras Justin Campbell 2011-10-04T02:36:43Z 2011-10-04T02:36:43Z <p>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.</p> <p>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 <a href="http://www.ams.org/journals/bull/1999-36-04/S0273-0979-99-00849-6/home.html" rel="nofollow">here</a>.</p>