# List of Casimir elements of low dimensional Lie algebras

I am interested in explicit formulae for the Casimir elements (or "Casimir operators") of low-dimensional, real, non-Abelian Lie algebras (d=2,3, and possibly 4). I am wondering if there is any reference with a list of, at least some, expressions for such Casimir elements. Even a few explicit expressions for some simple examples would be highly appreciated.

I know that there exist formal definitions and procedures, but, since those low-dimensional algebras are classified (e.g. https://en.wikipedia.org/wiki/Classification_of_low-dimensional_real_Lie_algebras) and well studied, I would like to avoid the whole construction of the enveloping algebra and the Casimir elements by hand. However, I could not find explicit formulae in the literature, since most books and lecture notes only give the general recipe and provide the typical example of $$\mathfrak{su} (2)$$.

Of course, if not a unique "list", also some references where I could retrieve different formulae for different cases would be a great answer!

Since I am a physicist, I apologise if my notation is unclear or imprecise. Thank you all in advance for any help.

• The Casimir element means several things (for a representation? in the universal enveloping algebra?) but in all cases it makes use of a non-degenerate symmetric bilinear form, usually the Killing form. In particular I have no idea how you could define a Casimir element of the Heisenberg Lie algebra or in the 2-dimensional non-abelian Lie algebra. – YCor Jul 23 at 11:45
• I'm not sure how you are defining rank for the Heisenberg Lie algebra, but the standard definition I know, says the rank is 2 and hence there is one invariant function: what you call $z$. The number of independent invariants is the co-rank (= dimension - rank). – José Figueroa-O'Farrill Jul 23 at 13:40
• Thank you for you comment @JoséFigueroa-O'Farrill, it has been very useful: I was mistaken in the number of invariants (I though "=rank", instead of "=co-rank"). I edited the question consistently, removing the misleading example. – Aureliano Tinajero Jul 23 at 14:43
• @JoséFigueroa-O'Farrill I don't think there's a single "standard" definition of rank for finite-dimensional, say complex, Lie algebras. Actually I know at least 2, and I'm not sure at all you have in mind one of those: (a) the dimension of some/every Cartan subalgebra (this is the usual rank in the semisimple case, and the dimension in the nilpotent case); (b) the dimension of a maximal torus in the automorphism group (again this the usual rank in the semisimple case; in the nilpotent case it's between $0$ and the dimension of the abelianization). – YCor Jul 23 at 16:07
• @YCor I agree that there are many notions of "rank", but in the present context: namely, that of determining the number of invariant functions on a Lie algebra, I believe the standard notion of rank is the one in the paper of Patera, Sharp and Winternitz mentioned in the answer by Zurab Silagadze. – José Figueroa-O'Farrill Jul 24 at 10:18