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Given a general Lie algebra, is there a general procedure to find all its Casimir operator?

flag	asked Aug 27 2011 at 11:11 cyl 21●2

If the Lie algebra is semisimple, then the center of its universal enveloping algebra is polynomial, and lists of good generators (often "good" means "homogeneous with respect to some grading") are known. In general, I think there is the Duflo theorem that $Z(U(\mathfrak g)) = \operatorname{Sym}(\mathfrak g)^{\mathfrak g}$ as rings, but the map is somewhat nontrivial. And anyway, this just moves the problem: certainly I don't know how, for a general Lie algebra, to compute the ring structure on $\operatorname{Sym}(\mathfrak g)^{\mathfrak g}$ (GIT quotient of adjoint action). – Theo Johnson-Freyd Aug 27 2011 at 14:08

Igor Rivin · Answer 1 · 2011-08-27 14:02:53Z UTC

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http://www2.ucy.ac.cy/~symmetry/Talks08/Boyko.pdf

answered Aug 27 2011 at 14:02

Igor Rivin
31.2k●3●37●71

The method in this paper is particularly suited for solvable Lie algebras, though, as it requires an explicit parametrisation of the adjoint group. Of course, the OP asks about "a general Lie algebra" so this goes some way towards answering the question. +1 – José Figueroa-O'Farrill Aug 27 2011 at 17:12

H. Arponen · Answer 2 · 2011-09-08 11:28:59Z UTC

I'm assuming you're thinking of some specific matrix representation $X_i \in \mathfrak{g}$ (let's assume it's the defining representation). Compute the Killing form, $\kappa_{ij} \doteq Tr (X_i\cdot X_j)$ (actually usually this is defined in the adjoint representation, but any faithful rep will do). The quadratic Casimir is then simply $ X_i \kappa_{ij} X_j$ (Einstein convention).

Other Casimirs can be obtained from the characteristic (secular) equation: define $X(\omega) = \omega^i X_i$. The characteristic equation is $\det\left( X(\omega) - \lambda I \right) = \sum\limits_{j} (-\lambda)^{N-j} \phi_j(\omega) \equiv 0 $ ($N$ is the matrix dimension, and/or the dimension of the Lie algebra if you're using the adjoint representation). If you now perform the substitution $\omega^i \to X_i$ in the coefficients $\phi_j (\omega)$, you get Casimir invariants $\phi_j (X)$!

It might seem like the higher the representation, the more invariants you get, but in fact all the invariants can be expressed in terms of the fundamental invariants of the defining representation. I can't think of many references right at the moment, but e.g. Gilmore: Lie groups, physics and geometry pp. 140 has a nice explanation. Also, google the boldface texts above.

How to find Casimir operators?

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