# How to find Casimir operators?

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

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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 '11 at 14:08

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 $\phi_j (X)$!