This question is prompted by a recent MO question on explicit computations of Weyl group invariants for certain exceptional simple Lie algebras: 37602. Like some others who started graduate study in the 1960s with almost no physics background but with an interest in abstract mathematics, I was drawn to algebraic Lie theory for mainly esthetic reasons. I also had no background in differential geometry or Lie groups. So when I bought a copy of Jacobson's newly-published book on Lie algebras at a bookstore in Ithaca I had no appreciation of the historical connections of the subject.

The eminent Dutch physicist H.B.G. Casimir was apparently the first to introduce an explicit second degree invariant (unique up to scalars) in the center of $U(\mathfrak{g})$, now called the Casimir element or Casimir invariant. Roughly speaking, this involves fixing a basis of $\mathfrak{g}$ along with its dual basis under the Killing form, then adding the respective products. Sometimes it is convenient to recast the answer in terms of PBW monomials for the given ordered basis.

On the mathematical side, Chevalley and Harish-Chandra determined the full center of $U(\mathfrak{g})$: it is a polynomial algebra in $\ell$(= rank of $\mathfrak{g}$) variables. Generators can be taken to be homogeneous of uniquely determined degrees. Moreover, the center is isomorphic in a natural way (but requiring a subtle $\rho$-twist) to the algebra $U(\mathfrak{h})^W$ of invariants of the Weyl group relative to a fixed Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$. The earlier MO question involved this algebra and its (non-unique) homogeneous generators of degrees $2 = d_1, d_2, \ldots, d_\ell$. Key papers were those by Harish-Chandra (Trans. AMS 1951) and Chevalley (Amer. J. Math. 1955), the latter generalizing the fundamental theorem on elementary symmetric polynomials for $W=S_n$.

My question then is:

What role did Casimir's work play in this mathematical development?

A related matter is the practice of referring to homogeneous generators of the center of the enveloping algebra as "Casimir operators": how far was Casimir himself involved in this direction beyond his degree 2 invariant?

ADDED: The short article referenced by mathphysicist is illuminating and may be the best published indication of Casimir's influence on subsequent representation theory. I was at first hoping to find a more definite paper trail, but this may not exist and would probably reach back before Math Reviews. What struck me most in browsing through the first volume of Harish-Chandra's collected papers (1944-54) was the abrupt transition around 1948 from his work in physics like Motion of an electron to mathematics like Faithful representations of Lie algebras and of course his foundational 1951 paper I mentioned. Nowhere along the way do I see any direct citation of Casimir's papers, though the 1950 paper Lie algebras and the Tannaka duality theorem does quote the "Casimir operator" in rank 1 as well known and uses it as a stepping-stone to the general case. Since Harish-Chandra studied physics with Dirac in his early years, it was probably he who imported Casimir's idea into representation theory. But Chevalley was at the time also a major influence on Harish-Chandra's thinking, so it's all hard to document. (They both taught at Columbia for some time.)

  • $\begingroup$ Out of curiosity: your introduction to Lie theory was Jacobson's book? $\endgroup$ – Mariano Suárez-Álvarez Oct 5 '10 at 14:49
  • 2
    $\begingroup$ @Mariano: Yes. When I bought the book at Cornell I had just switched over to their math program from philosophy and was vaguely interested in algebra. After moving to Yale the next year I studied mainly algebraic groups, but took part in one seminar on Lie algebras with some faculty and students. Jacobson's was probably the only book on Lie algebras (in English) during that era, but Chevalley, Bourbaki, Serre were active in the late 1950s to 1960s (in French). In those days it was especially challenging to learn about Lie groups from books. $\endgroup$ – Jim Humphreys Oct 5 '10 at 16:25

At the first glance it appears that he more or less just gave the first nontrivial example(s) of what was later called the Casimir operators.

His obituary says:

On 1 May 1931 he wrote a letter from The Hague to the famous Gottingen mathematician Hermann Weyl and announced: ‘While studying the quantum-mechanical properties of the asymmetic rotator I arrived at some ‘results’ (?) concerning the representation of continuous groups.’ He then sketched his findings on the matrix elements of the irreducible representations for the three-dimensional rotation group, and a possible extension for semi-simple groups in general, where he introduced what was later called the ‘Casimir operator’. This operator turned out to be a multiple of the unit-operator and may be used to characterize in an elegant way the irreducible representations of a given continuous group. To Casimir’s question, ‘Whether the case is worth considering?’, Weyl answered definitely ‘Yes’. Hence the Leiden doctoral candidate published his mathematical results in a paper, communicated by Ehrenfest to the meeting of 27 June 1931 of the Amsterdam Academy [7], and he also included them as Chapter IV of his dissertation, which he defended on 2 November 1931 at the University of Leiden [8].

  • $\begingroup$ Thanks for pointing out this obituary from the European Journal of Physics 22 (2001), to which our library does have online access (a tricky issue for us in these days of relentless cuts to library budgets). I can see that Casimir was an early pioneer in recognizing the value in an important special case of what has become a standard tool in studying representation theory of semisimple groups or Lie algebras. $\endgroup$ – Jim Humphreys Oct 5 '10 at 19:44

Apologies if it isn't customary to answer questions asked and answered so long ago, I'm new here...

In his autobiography Het toeval van de werkelijkheid (1983), Casimir makes a short remark about his operator (there is an English translation, Haphazard reality, but I don't own it so this translation is my own):

My dissertation [...] also describes wat is nowadays often called the Casimir operator. To put this in the proper perspective, I would like to quote something from a witty English book called How to be famous. Concerning Plato, it says: 'His own inaccurate ideas about platonic love are outdated. In his own time, however, they were an improvement on the existing situation, where one had no concept of platonic friendship whatsoever.' Mutatis mutandis, the same could be said about Casimir operators.

This at least tells us he thought he was the first to find the Casimir operator, but he is very modest about his role in the general theory; he doesn't seem to have worked on it besides the case of the rotation group, and I think he says somewhere else that he feels it isn't fair that the operator is named after him.

So I'd agree (and so would Casimir himself): he was the first to find an example, but wasn't involved in the subsequent development.

  • $\begingroup$ Great find!${}{}$ $\endgroup$ – B R Apr 6 '12 at 2:03

A while back I was given a hardcopy of Casimir's thesis, and recently scanned it so that I could have an electronic version. The thesis doesn't appear to be readily available online, so thought it would be worth sharing here. You can download it from my public Dropbox here. The quadratic Casimir is introduced in Theorem III on page 93 of the thesis (page 52 of the scan) - the $\mathcal{D}_\mu$ are elements of the Lie algebra, considered as (right-invariant, I think) differential operators on $C^\infty(G)$, and $g^{\lambda\mu}$ is the inverse of the Killing metric with respect to this basis. Casimir actually considered the case of an arbitrary semi-simple Lie group, and not just the rotation group.

EDIT: adding a Scribd link here for those who would prefer not to download the entire 25 meg file.

  • $\begingroup$ Great, nice to have this! I like how he consistently writes 'selvadjoint'. $\endgroup$ – Philip van Reeuwijk Apr 11 '12 at 14:31
  • $\begingroup$ @Paul: Thanks for this documentation, even though I haven't quite mastered Dutch yet (beyond 'selvadjoint'). It's interesting to confirm that Casimir did look at differential operators for arbitrary semi-simple Lie groups even if that didn't immediately influence the physics literature. $\endgroup$ – Jim Humphreys Apr 11 '12 at 16:36
  • $\begingroup$ @Jim: Surprisingly most of the thesis is in English (with the occasional compound "fourdimensional" and "matrixelements" scattered here and there). So don't let your lack of Dutch put you off taking a look :) $\endgroup$ – user17945 Apr 11 '12 at 17:12
  • $\begingroup$ @Paul: My comment was a bit facetious, since my real difficulty is to relate the physics context to the "pure" mathematics surrounding representation theory and Lie theory. But the thesis does provide good historical documentation for the early thinking of Casimir. This is mostly lost by now in the ciurrent literature where his name is used. $\endgroup$ – Jim Humphreys Apr 11 '12 at 19:59

One could say that Casimir “imported his idea into representation theory” himself, jointly with van der Waerden, when they used it to prove complete reducibility in rank $l>1$ (1935; his Festschrift quotes, p. 114, an interesting letter of Pauli to Weyl on how this came about).

But I get that the title question is different. On that, Borel (1998, p. 79) attributes to Racah (1950) the idea of further simplifying the proof by using $l$ independent Casimirs instead of just one. (Both papers have notable reviewers.) He says that Chevalley and Harish-Chandra had totally different motivations. However: H.-C.’s obituary (1985, p. 208) cites a 1951 letter where he “reports on the work of G. Racah on the invariants of the exceptional groups, with its application to the calculation of their Betti numbers”.

(Does the $\mathfrak Z(\mathfrak g) = U(\mathfrak h)^W$ picture allow a simplified proof of complete reducibility?)


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