# Terminology for nilpotent groups

I have a nilpotent lie group $N$ with upper central series $$1 = N_0 \triangleleft N_1 \triangleleft \dots \triangleleft N_k = N$$ which induces the filtration $$0 = \mathfrak{n}_0 \subset \mathfrak{n}_1 \subset \dots \subset \mathfrak{n}_k = \mathfrak{n}$$ of the Lie algebra $\mathfrak{n}$.

For convenience, I've defined the level of a vector $\xi \in \mathfrak{n}$ to be the smallest $i$ such that $\xi \in \mathfrak{n}_i$. 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?

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I would call it the "nilpotent order", as I usually call "order" of an element the index of smallest layer of the filtration which contains it, because I like differential operators. –  Mariano Suárez-Alvarez May 24 '10 at 15:40
Certainly "level" is going to conflict with many other notions. If MSA's suggestion "order" is unavailable, I'd maybe use "length" or "power". –  Theo Johnson-Freyd May 24 '10 at 23:21
@Theo: All of these standard single words seem ambiguous in this specialized context. The closest might be "degree" if interpreted as "graded degree" in the associated grading. Otherwise it's probably clearer just to write $x \in \mathfrak{n}_i \setminus \mathfrak{n}_{i-1}$. –  Jim Humphreys May 25 '10 at 14:56
P.S. Using a familiar word in this situation is tricky, but it's harmless for repeated reference to say something like "Write $d(x) :=i$" without making up a word. –  Jim Humphreys May 25 '10 at 16:41

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, Groupes et algebres de Lie), 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 $\mathfrak{n}_i$, it's typical just to refer to an element $x$ lying in $\mathfrak{n}_i$ but not in the previous term. Perhaps somewhere in the literature a term like level or height or whatever might get used to refer to the index $i$ 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 filtration level would be safe when an ascending filtration is fixed.
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.