$\mathfrak{sl}_2(\mathbb{C})$ is usually given a basis $H, X, Y$ satisfying $[H, X] = 2X, [H, Y] = 2Y, [X, Y] = H$. What is the origin of the use of the letter $H$? (It certainly doesn't stand for "Cartan.") My guess, based on similarities between these commutator relations and ones I have seen mentioned when people talk about physics, is that $H$ stands for "Hamiltonian." Is this true? Even if it's not, is there a connection?
What's the origin of the naming convention for the standard basis of $\mathfrak{sl}_2(\mathbb{C}) $?

23$\begingroup$ For me, it is usually named $E$, $F$, $H$... $\endgroup$– Mariano SuárezÁlvarezMar 11, 2010 at 4:19

2$\begingroup$ X, Y, H is the convention used by the Wikipedia article and by Fulton and Harris. I hope the answer isn't just that H is the next letter after G... $\endgroup$– Qiaochu YuanMar 11, 2010 at 5:05

6$\begingroup$ The connection to Hamiltonians seems unlikely to me. As the maximal torus of G is a subgroup, calling it H and the corresponding Lie algebra element h is quite natural. I hope someone can pinpoint the etymology though. $\endgroup$– Q.Q.J.Mar 11, 2010 at 5:25

4$\begingroup$ I had always imagined that H was called H because Cartan subalgebras are always called mathfrak{h}. Of course I now realise that it could have been the other way around! Which convention came first I wonder? $\endgroup$– Kevin BuzzardMar 11, 2010 at 11:43

2$\begingroup$ Well, at least we can be sure this h wasn't due to Dirichlet. (anyone who doesn't catch the joke should look at mathoverflow.net/questions/17062/…) $\endgroup$– KConradMar 11, 2010 at 14:54
3 Answers
The letters $\mathrm X$ and $\mathrm Y$ are already used by Cayley in what Dieudonné (in MR) calls the first description of all finitedimensional irreducible $\mathfrak{sl}_2$modules: A Second Memoir upon Quantics (1856, §§29–31). He apparently has no name for $\mathrm{XYYX}$. Same in e.g. Faà di Bruno (1876, §§113–114).
$\mathrm H$ seems to stand for Hauptmatrix, as introduced by Weyl (1922, p. 125; see also 1931, pp. 114, 122) to describe Cartan subalgebras, roots (Multiplikatoren) and root spaces (Länder):
Kommt in $\mathfrak g$ eine „Hauptmatrix“ $H$ vor, in der alle Elemente außerhalb der Hauptdiagonale verschwinden, während in der Hauptdiagonale die Zahlen $\alpha_1,\alpha_2,\dots,\alpha_n$ stehen, so bilde man die Differenzen $\alpha_i  \alpha_k$ und teile mit Bezug auf $H$ das Schema einer beliebigen Matrix in „Länder“ ein, indem man jedem Feld $(ik)$ des Schemas ($i$ der Zeilen, $k$ der Kolonnenindex) die Zahl $\alpha_i  \alpha_k$ als „Multiplikator“ zuordnet (...)
$\mathfrak{sl}_2$triples with your bracket relations appear in Killing (1888, p. 281), denoted $(X_{r1}, X_r, X_{r2})$; Cartan (1894, p. 116), denoted $\mathrm{(Y, X,X')}$; Weyl (1925, p. 276), denoted $(h_\alpha,e_\alpha,e_{\alpha})$, with the $h_\alpha$ again called “Diagonal oder Hauptmatrizen”; Dynkin (1952, §8.1), denoted $(f,e_+,e_)$; Chevalley (1955, p. 28; 1955, p. 96), denoted $(D,N,N')$, $(H_r,X_r,X_{r})$, and finally the desired $\mathrm{(H,X,Y)}$.
(Standardization was slow: Lie (1876, p. 53; 1890, p. 353) used $(X_1,X_2,X_3)=(\mathrm{Y,\smash{\frac12}H,X})$, and similarly rescaled bases and bracket relations still appear in Pauli (1927, p. 614), BornJordan (1930, p. 135), Casimirvan der Waerden (1931, p. 46; 1935, p. 4), Bauer (1933, p. 126), HarishChandra (1950, p. 301; 1952, p. 337), Jacobson (1951, p. 107; 1958, p. 825), Séminaire “Sophus Lie” (1955, p. 1001), Kostant (1959, p. 977), etc. Settling on the “Chevalley” basis $\mathrm{(H,X,Y)}$ over Lie’s seems ultimately motivated by the smaller $\mathbf Z$form $\mathfrak g_\mathrm{sc}\subset\mathfrak g_\mathrm{ad}$ it spans, cf. Borel (1970, §2.7).)

$\begingroup$ In Killing's indexing, what does the $r$ stand for? The notation suggests that maybe I have triples $(X_0, X_1, X_2)$ and $(X_3, X_4, X_5)$ or so; or is that not how it works? $\endgroup$– LSpiceJul 29, 2019 at 14:03

1$\begingroup$ @LSpice Killing has $r=\dim\mathfrak g$ throughout; in §8, he seems to be dealing with a semisimple $\mathfrak g$ having an $\mathfrak{sl}_2$ factor, which he takes to be spanned by the last 3 basis vectors (so e.g., $X_1$, $X_2$, $X_3$ when $\mathfrak g$ is $\mathfrak{sl}_2$). $\endgroup$ Jul 30, 2019 at 0:40

1$\begingroup$ Thank you! Cool to see this finally get answered after so long. $\endgroup$ Jul 31, 2019 at 17:28
Both terminology and notation in Lie theory have varied over time, but as far as I know the letter H comes up naturally (in various fonts) as the next letter after G in the early development of Lie groups. Lie algebras came later, being viewed initially as "infinitesimal groups" and having labels like those of the groups but in lower case Fraktur. Many traditional names are not quite right: "Cartan subalgebras" and such arose in work of Killing, while the "Killing form" seems due to Cartan (as explained by Armand Borel, who accidentally introduced the terminology). It would take a lot of work to track the history of all these things. In his book, Thomas Hawkins is more concerned about the history of ideas. Anyway, both (h,e,f) and (h,x,y) are widely used for the standard basis of a 3dimensional simple Lie algebra, but I don't know where either of these first occurred. Certainly h belongs to a Cartan subalgebra.
My own unanswered question along these lines is the source of the now standard lower case Greek letter rho for the halfsum of positive roots (or sum of fundamental weights). There was some competition from delta, but other kinds of symbols were also used by Weyl, HarishChandra, ....
ADDED: Looking back as far as Weyl's fourpart paper in Mathematische Zeitschrift (19251926), but not as far back as E. Cartan's Paris thesis, I can see clearly in part III the prominence of the infinitesimal "subgroup" $\mathfrak{h}$ in the structure theory of infinitesimal groups which he laid out there following Lie, Engel, Cartan. (Part IV treats his character formula using integration over what we would call the compact real form of the semisimple Lie group in the background. But part III covers essentially the Lie algebra structure.) The development starts with a solvable subgroup $\mathfrak{h}$ and its "roots" in a Fitting decomposition of a general Lie algebra, followed by Cartan's criterion for semisimplicity and then the more familiar root space decomposition. Roots start out more generally as "roots" of the characteristic polynomial of a "regular" element. Jacobson follows a similar line in his 1962 book, reflecting his early visit at IAS and the lecture notes he wrote there inspired by Weyl.
In Weyl you also see an equation of the type $[h e_\alpha] = \alpha \cdot e_\alpha$, though his notation is quite variable in different places and sometimes favors lower case, sometimes upper case for similar objects. Early in the papers you see an infinitesimal group $\mathfrak{a}$ with subgroup $\mathfrak{b}$. All of which confirms my original view that H is just the letter of the alphabet following G, as often encountered in modern group theory. (No mystery.)

1$\begingroup$ Pretty sure Knapp uses $\delta$ for the halfsum. The real showdown is between $\Pi$ and $\Delta$ or $\Delta$ and $R$ for the simple roots and all roots. $\endgroup$– Q.Q.J.Mar 13, 2010 at 8:18

$\begingroup$ Probably I picked up the use of
$\delta$
from Jacobson's book on Lie algebras, but conventions have varied for a long time. The names of root systems or simple roots also vary from one source to another. After 1968 Bourbaki standardized usage somewhat, but I think made a mistake in switching to roman letters$R$
and$B$
(the former common in English for rings and the latter ubiquitous for Borel subgroups). I've pretty much followed Borel and Tits, using$\Phi$
and$\Delta$
(the latter often being replaced by a numbered list of simple roots). $\endgroup$ Mar 22, 2010 at 17:09
I always thought that $H$ did stand for Cartan—at least, for ‘Henri’. However, I seem to recall that I had this discussion with Brian Conrad, and that he said it was actually Élie, not Henri, after whom the subgroups were named.
For what it's worth, the $(X, Y, H)$ convention (in preference to $(E, F, H)$) is the one to which I'm accustomed; Carter uses it, for example, in his discussion of Jacobson–Morosov triples. Embarrassingly, I don't know where the Jacobson–Morosov theorem was proven; but that's where I'd look for the history of the name.

$\begingroup$ Following Lie and Killing, Elie Cartan (father of Henri) laid foundations for Lie groups and "infinitesimal" groups (later Lie algebras): Paris thesis 1894, etc. Killing studied "Cartan" subalgebras first but didn't invent the "Killing" form. Even in Bourbaki, notation varies:
$(H, X_+, X_1), (X,Y,H), (x,h,y)$
. JacobsonMorozov comes from separate work (Duke J. 1938?, Doklady note 1942): see Jacobson (1962 book) pp. 98100, where$(x,h,y)$
is used. Upper case is common for Lie algebra elements viewed as vector fields or as matrices. Lots of obscure history, but does it matter? $\endgroup$ Mar 31, 2010 at 13:31 
$\begingroup$ Hmmm, my math got distorted in the previous comment. Also, I should have typed
$(H,X_+, X_)$
. This kind of notation has been fairly popular with physicists but also occurs in Bourbaki. $\endgroup$ Mar 31, 2010 at 16:29 
$\begingroup$ Jim, thanks for the history, and especially for the attribution! I'm not sure what to make of the "does it matter?" question—I guess it matters to Qiaochu, and certainly I find it interesting. $\endgroup$– LSpiceMar 31, 2010 at 17:05