Given a general Lie algebra, is there a general procedure to find all its Casimir operator?
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3$\begingroup$ 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). $\endgroup$– Theo Johnson-FreydAug 27, 2011 at 14:08
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2 Answers
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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 $\omega^i \mapsto X_i$ 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.
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1$\begingroup$ I've opened a new post regarding a concrete application of the recipe that you've explained. I hope that you will take a look. Thanks in advance. mathoverflow.net/questions/312768/… $\endgroup$ Oct 14, 2018 at 9:17
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1$\begingroup$ wow blast from the past! Sorry but I haven't dealt with this stuff in a long time... anyway the answer by Vit Tucek in the post you linked seems pretty good at a glance $\endgroup$ Oct 20, 2018 at 19:41
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$\begingroup$ Ah ok! I saw that the post was put 7 years ago and I imagined that in the meanwhile maybe you've changed field. Anyway thanks for your comment! $\endgroup$ Oct 21, 2018 at 9:57
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https://arxiv.org/abs/math-ph/0602046
Computation of Invariants of Lie Algebras by Means of Moving Frames -Vyacheslav Boyko, Jiri Patera, Roman Popovych
answered Aug 27, 2011 at 14:02
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$\begingroup$ 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 $\endgroup$ Aug 27, 2011 at 17:12
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4$\begingroup$ Due to "link rot", this answer is now useless :( Could anyone reproduce the author(s) & title of the linked article? $\endgroup$– DanuFeb 10, 2016 at 11:14
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2$\begingroup$ @ChanBae Some version of the link you removed can still be found using the Wayback Machine. The link seems to be a PDF-file with a talk related to the paper you linked - so the arXiv link seems to be the correct paper. (And naturally, I should at least say: Thanks for your help with replacing the dead links!) $\endgroup$ Jun 24, 2022 at 5:05