Talk:Casimir invariant

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Mathematics rating:	Start Class	Mid Priority	Field: Algebra
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Geometry guy 22:23, 16 June 2007 (UTC)

need to disambiguate rank. there is currently no page for the rank of a Lie algebra/Lie group. does this deserve its own page? or is it better explained in the Lie algebra and Lie group articles?Lethe 02:04, 9 Mar 2004 (UTC)

[edit] Formulation

Added the precise formulation for the case of semisimple Lie algebras, but unsure what modification is required in other possible cases - Most sources (Knapp, Helgason, & Jacobson) appear to only consider the semisimple case. Dmaher 01:09, 8 July 2006 (UTC)

[edit] Cleanup & Expert tags

I have added "cleanup" and "expert" tags to the article for the following reasons. Currently (16:24, 11 July 2006 (UTC)) what the article discusses, and the introduction defines, is just a quadratic (second-order) Casimir operator. This is just one of many possible Casimir operators, although perhaps the most common and best known (especially in physical applications). General Lie algebras can possess other, higher-dimensional Casimir operators. A more general definition should be given by someone who is an expert on this topic. 131.111.8.98 16:24, 11 July 2006 (UTC)

Well, one thing is definitely missing: the article completely sidesteps the issue where do these operators actually "live" — it should be the (center of) the universal enveloping algebra, although of course, they induce an operator in any its representation. As for "higher Casimir operators", this terminology is rarely used in representation theory, at least within mathematics. I can think of only one major book, Zhelobenko's "Compact Lie groups and their representations", AMS Translations, vol 40, 1973, which employs this terminology. I know that the convention is different in physics literature, where the term "Casimir operator" may refer to more general elements of the center of the universal enveloping algebra. In that case, though, they are called "Higher Casimir elements" or something similar. I think the problem partly arises from the following incorrect sentence in the universal enveloping algebra article:

Under this representation, the elements of U(L) invariant under the action of L (i.e. such that any element of L acting on them gives zero) are called invariant elements. They are generated by the Casimir invariants.

Arcfrk 11:32, 10 March 2007 (UTC)

The usage "higher", while perhaps rare in mathematics, is common in physics, and is used to distinguish the quadratic case from more complex invariants, e.g. on supersymmetric manifolds, etc.

Anyway, I'm removing the "needs expert attention" tags; by the criteria given, 98% of all math articles on WP need "expert attention".

FWIW, my PhD thesis had the words "Casimir operator" in its subtitle, and I would be rather hard-pressed to give a general definition that corresponds to the Lie-algebra definition. In my case, it was built from a Dirac operator, integrated over a volume. It was a cocycle (i.e. vanishing derivative), and so corresponded to a conserved current, by Noether's theorem. The "conserved charge" of this current was the Casimir invariant. I still don't know how to turn this into a general definition. linas 13:55, 15 May 2007 (UTC)