Terminology for nilpotent groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T23:41:47Z http://mathoverflow.net/feeds/question/25777 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25777/terminology-for-nilpotent-groups Terminology for nilpotent groups Matt Noonan 2010-05-24T15:37:22Z 2010-05-25T07:22:27Z <p>I have a nilpotent lie group $N$ with upper central series <code>$$1 = N_0 \triangleleft N_1 \triangleleft \dots \triangleleft N_k = N$$</code> which induces the filtration <code>$$0 = \mathfrak{n}_0 \subset \mathfrak{n}_1 \subset \dots \subset \mathfrak{n}_k = \mathfrak{n}$$</code> of the Lie algebra $\mathfrak{n}$. </p> <p>For convenience, I've defined the <em>level</em> of a vector $\xi \in \mathfrak{n}$ to be the smallest $i$ such that <code>$\xi \in \mathfrak{n}_i$</code>. I assume this concept has a standardized name, either in the context of filtrations or in the context of nilpotent Lie algebras. What would this standardized name be?</p> http://mathoverflow.net/questions/25777/terminology-for-nilpotent-groups/25788#25788 Answer by Jim Humphreys for Terminology for nilpotent groups Jim Humphreys 2010-05-24T17:38:52Z 2010-05-24T17:38:52Z <p>I doubt very much that a "standardized" name for the concept exists in the context of nilpotent Lie algebras, to judge by a quick look at older books by Bourbaki (Chapter 1, 1960, <em>Groupes et algebres de Lie</em>), Jacobson, Dixmier. I'm less familiar with terminology in the theory of Lie rings and abstract nilpotent groups. Once the terms of the upper (or ascending) central series are labelled as something like <code>$\mathfrak{n}_i$</code>, it's typical just to refer to an element <code>$x$</code> lying in <code>$\mathfrak{n}_i$</code> but not in the previous term. Perhaps somewhere in the literature a term like <em>level</em> or <em>height</em> or whatever might get used to refer to the index <code>$i$</code> here, but such terms tend to be used for other purposes in Lie theory (I've even seen a reference to "level of a nilpotent Lie algebra" in the context of varieties of Lie algebras). Probably <em>filtration level</em> would be safe when an ascending filtration is fixed. </p> http://mathoverflow.net/questions/25777/terminology-for-nilpotent-groups/25852#25852 Answer by Yiftach Barnea for Terminology for nilpotent groups Yiftach Barnea 2010-05-25T07:22:27Z 2010-05-25T07:22:27Z <p>I am not sure that there is a standard answer, so you will need to define it. But in profinite group theory we tend to call it degree, (if you have a pro-$p$ group and nice enough filtration, you can associate a graded Lie algebra and elements in the group map to homogenous elements in the Lie algebra, so degree makes sense), although, we usually deal with descending filtrations.</p>