# Translation of “le nilradicalisé de g”

I apologize for asking something that might well be found in a mathematical dictionary, but the similarity of the French word to an English one is frustrating my attempts to Google the answer (and the library is shut at time of typing). I suspect the answer should be obvious to those who, unlike me, know some basic Lie group/Lie algebra terminology.

Some context: I am reading an old paper of Dixmier from 1969, which has the following construction/definition. Let $\mathfrak g$ be a Lie algebra (characteristic zero, finite-dimensional), let $\mathfrak n$ be its largest nilpotent ideal -- the nilradical -- and put ${\mathfrak h}=[{\mathfrak g},{\mathfrak g}]+{\mathfrak n}$. Dixmier calls ${\mathfrak h}$ "le nilradicalisé de ${\mathfrak g}$".

Literal translation would surely be "the nilradicalised", but that sounds more like a mopey university indie band than a mathematical object. So what is the usual name for this object in English?

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I don't know the English term for the object, but let me assure you that "the nilradicalized" actually sounds better than "le nilradicalisé"! (Given the year 1969, this could very well be a joke.) –  François G. Dorais Jan 31 '10 at 23:10
The thought of Dixmier making jokes somehow doesn't compute in my head -- but I only know (some of) his work, so perhaps I'm selling him short. –  Yemon Choi Jan 31 '10 at 23:28

## 1 Answer

I suspect pretty strongly that this is idiosyncratic terminology; I've never seen that subalgebra used, and the term has no google hits other than this post.

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Thanks for confirming my suspicion. The paper's real goal is to say something about the von Neumann algebra of a connected Lie group; but I wasn't sure if this preliminary construction was somehow standard. –  Yemon Choi Jan 31 '10 at 23:55
I've asked my dept neighbour who is a "Lie algebras person" and he too has neither heard the term nor seen that subalgebra. So it looks like Ben's answer is as good as we'll get. –  Yemon Choi Feb 3 '10 at 3:49
Something I may have noticed at the time but forgotten: MathSciNet gives exactly one match for the term "nilradicalis&eaacute;", and it is in Kleppner's review of a Trans. AMS paper of Pukanszky ams.org/mathscinet-getitem?mr=486313 (The definition is made in the paper on page 22, just before Lemma 7; but Pukanszky never uses the term nilradicalis&eacute;) –  Yemon Choi Jan 5 '13 at 0:10