Talk:Casimir invariant

From Wikipedia, the free encyclopedia

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)