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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.

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    $\begingroup$ 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. $\endgroup$
    – YCor
    Commented Jul 23, 2019 at 11:45
  • $\begingroup$ 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). $\endgroup$
    – José Figueroa-O'Farrill
    Commented Jul 23, 2019 at 13:40
  • $\begingroup$ 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. $\endgroup$
    – Aureliano Tinajero
    Commented Jul 23, 2019 at 14:43
  • $\begingroup$ @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). $\endgroup$
    – YCor
    Commented Jul 23, 2019 at 16:07
  • $\begingroup$ @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. $\endgroup$
    – José Figueroa-O'Farrill
    Commented Jul 24, 2019 at 10:18

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A more general problem (finding all invariant functions of the group generators for all real Lie algebras of dimension up to five) was considered in Invariants of real low dimension Lie algebras by J. Patera, R. T. Sharp, and P. Winternitz. See also a more recent paper Computation of Invariants of Lie Algebras by Means of Moving Frames by V. Boyko, J. Patera and R. Popovych.

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  • $\begingroup$ Thank you, those references are extremely useful! $\endgroup$
    – Aureliano Tinajero
    Commented Jul 23, 2019 at 15:31

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