# Casimir operators of a given Lie Algebra

I am a Physicist, so let me apologize in advance for some possible imprecisions.

I'm working on a 10-dimensional Lie Algebra. Each element of the algebra represents a quantum mechanical operator, and I would like to find the "constants of motions" of my system. These, in turn, correspond to the Casimir operators of the algebra.

First of all: the number of Casimir operators corresponds to the rank of the Algebra, doesn't it? And the rank of the algebra is the maximum number of operators that commute among themselves, isn't it?

I've already written the matrix realization of my algebra and now I'm looking for an "algorithmic" way of findining the Casimir operators. I know my question sounds similar to this one: How to find Casimir operators?, yet I haven't understood it fully. Can you please give a detailed and practical guide on how to find them? If you wish, you can use the easy algebra $\mathbf{su}(2)$, pretending not to already know the final result.

• The answer clearly depends on the specific Lie algebra you consider. What is it, in standard mathematical or physical nomenclature? By the way, usually the term "rank" is reserved for semisimple Lie algebras (an important but very special class of Lie algebras). In general, one speaks of index, say $\ell$; however, it is no longer true that the invariant operators are freely generated by $\ell$ (algebraically) independent "Casimir elements". – Victor Protsak Feb 22 '17 at 4:30
• Also, the "maximum number of commuting elements" presumably corresponds to the size of a maximal commutative subalgebra of $U(L)$, the universal enveloping algebra of the Lie algebra $L$. This is typically much larger than the size of the center of $U(L)$, which consists of all operators that commute with everything in $U(L)$, and not just with each other. – Victor Protsak Feb 22 '17 at 4:38
• I think that you would benefit from reading one of many books on Lie groups for physicists. Typically, these books present explicit calculations of Casimir elements and you may be able to find the answer in your specific case. – Victor Protsak Feb 22 '17 at 4:41
• Thanks for the clarifications. Actually I was looking for a general method, able to find the Casimirs operators despite the particular nomenclature. In my case the algebra should be something like $\mathbf{so}(3,2)$. Please take a look to the second answer (the one by H.Arponen) at this post: mathoverflow.net/questions/73836/how-to-find-casimir-operators . I guess he is talking about what I'am looking for. He introduces the Killing form, etc... Maybe you could resume his answer and better clarify it. – AndreaPaco Feb 22 '17 at 8:50