TITLE: What's the origin of the naming convention for the standard basis of $\mathfrak{sl}_2(\mathbb{C}) $?
QUESTION [15 upvotes]: $\mathfrak{sl}_2(\mathbb{C})$ 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?

REPLY [16 votes]: The letters $\mathrm X$ and $\mathrm Y$ are already used by Cayley in what Dieudonné (in MR) calls the first description of all finite-dimensional irreducible $\mathfrak{sl}_2$-modules: A Second Memoir upon Quantics (1856, §§29–31). He apparently has no name for $\mathrm{XY-YX}$. 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 $(X_{r-1}, X_r, X_{r-2})$; 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), Born-Jordan (1930, p. 135), Casimir-van der Waerden (1931, p. 46; 1935, p. 4), Bauer (1933, p. 126), Harish-Chandra (1950, p. 301; 1952, p. 337), Jacobson (1951, p. 107; 1958, p. 825), Séminaire “Sophus Lie” (1955, p. 10-01), 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).)