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Consider the quite general problem of computing all Casimir Operators of a given Lie Algebra $\mathfrak{g}$. How does one proceed, in general? And how is possible to compute the degree of a given Casimir Operator?

I only know how to find the quadratic one, using a physical analogy with the square of the 4-momentum, or the angular momentum.

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  • $\begingroup$ What dou you mean by "computing all'' operators? It is unique... Also, Casimir operator is defined on semisimple, not arbitrary algebras. What do you mean by ``quadratic'' Casimir? $\endgroup$
    – user68061
    Feb 22, 2014 at 20:27
  • $\begingroup$ Fine for the remark on the fact it is only defined for semisimple algebras. However, I am sure you are wrong on the fact it is unique: it is not, since there are $r$ different casimir operators, where $r$ is the rank of the group. With quadratic I mean that it is built as the sum of products of two different generators. $\endgroup$ Feb 22, 2014 at 20:51
  • $\begingroup$ "casimir operator" can mean a lot of different things. If you are asking about an element of the center of $U(\mathfrak{g})$ then it is unique (this is the definition in wiki link). If you are talking about Casimirs of representations, then they are just the images of universal Casimir. Please specify your definition... $\endgroup$
    – user68061
    Feb 22, 2014 at 20:55
  • $\begingroup$ Im asking for the independent elements of the center of the universal enveloping algebra. Look at this en.wikipedia.org/wiki/Casimir_element. Section "Properties". Second paragraph. $\endgroup$ Feb 22, 2014 at 22:11
  • $\begingroup$ I understood, and answered accordingly... $\endgroup$
    – user68061
    Feb 26, 2014 at 12:30

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After clarification: you are not searching for the Casimir element, you are looking for the generators of the center. Then you will need Harish-Chandra isomorphism So your problem will be equivalent to finding a basis in the ring of "$W$-symmetric'' polynomials $S(\mathfrak{h})^W$'. This is a subject of invariant theory. As $W$ is finite the problem is not to hard, look for Noether finiteness theorem (e.g. in Craft-Procesi Invariant theory: A primer). Of course finding reasonable basis is a problem that depends on properties you want it to have. For the $A,B,C,D$ series you can find the disscussion in Humphreys "Reflection groups and Coxeter groups'' Section 3.12

P.S. The term "Casimir elements" for the generators is popular in physics, but never used in mathematics.

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