We know that $so(3)$ has the explicit quadratic Casimir

$$L^2=\sum L_{i}^2.$$

Are there analogs to this in other simple lie algebras?

I know that for a simple lie algebra I can always use the Cartan metric to obtain a formula for the quadratic Casimir; $$C=g^{ij}v_{i}v_{j}.$$

But then there is the problem of actually finding the Cartan metric, which (correct me if i'm wrong) can be troublesome if your algebra is high dimensional.

In particular, i'm interested in the quadratic Casimir of the simple Lie algebra $sp(2N)$. If anyone has a link where this Casimir is given explicitly (as is the Casimir for $so(3)$) I would appreciate it very much. Or even some insight into how one would get an explicit form for the cartan metric would be much obliged.

  • $\begingroup$ Explicit formula is given in the following possible duplicate: Definition of Casimir operator and its properties. $\endgroup$
    – Qmechanic
    Mar 19, 2018 at 19:32
  • $\begingroup$ Thanks for the heads up. I think this question may be a little different, because I just want to know if anyone has any nice explicit formula for the cartan metric. For instance the structure constants for so(3) are given by the totally antisymmetric symbol and then it's easy to find the cartan metric by contracting and using identities to be $\delta^{ij}$. I am wondering if such nice formulas are known in other lie algebras; in particular sp(2n). Thanks again $\endgroup$
    – S Thomas
    Mar 19, 2018 at 19:40
  • $\begingroup$ The resource you want is the review by Dick Slansky: Slansky, Richard. "Group theory for unified model building." Physics Reports 79.1 (1981): 1-128. available here: pdfs.semanticscholar.org/ca6d/… I dunno if it's there explicitly but if not it can be pretty easily constructed from the data in there. $\endgroup$ Mar 19, 2018 at 19:47
  • $\begingroup$ Thanks ZeroTheHero for the reference, I will check it out. @Qmechanic, I think what i'm looking for is still a little different. The possible duplicate has the definition of the cartan metric in terms of the structure constants. What i'm asking is if anyone has evaluated the definition of the cartan metric for any other specific simple lie algebras or classes of simple lie algebras to achieve a simple form the casimir. $\endgroup$
    – S Thomas
    Mar 19, 2018 at 20:04
  • $\begingroup$ If you only want the quadratic Casimir out of the N independent Casimirs of sp(2N), I recall Iachello's Lie Algebra book tells you how to put them together out of the "sensible" (dysfunctional) $E_{\alpha\beta}$ basis. The construction is reminiscent to that of the quadratic Casimir of so(2N+1) which has the same number of elements and rank and "shadows" sp(2N). Try sp(4) ~ so(5) to reassure yourself you paired the 10 generators properly. $\endgroup$ Mar 20, 2018 at 0:19


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