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In the paper The greatest mathematical paper of all time, by A. J. Coleman (The Mathematical Intelligencer 11 (1989) pp29-38, https://doi.org/10.1007/BF03025189) the author argues that Wilhelm Killing's 1889 paper Die Zusammensetzung der stetigen, endlichen Transformationsgruppen. Zweiter Theil (Mathematische Annalen 33 (1889) pp1-48 https://eudml.org/doc/157397), referred as Z.v.G.II, was the most significant mathematical paper he have read or heard about in fifty years, because:

(1) It was the paradigm for subsequent efforts to classify the possible structures for any mathematical object. Hawkins [15] documents the fact that Killing's paper was the immediate inspiration for the work of Cartan, Molien, and Maschke on the structure of linear associative algebras which culminated in Wedderburn's theorems. Killing's success was certainly an example which gave Richard Brauer the will to persist in the attempt to classify simple groups.

(2) Weyl's theory of the representation of semi-simple Lie groups would have been impossible without ideas, results, and methods originated by Killing in Z.v.G.II. Weyl's fusion of global and local analysis laid the basis for the work of Harish-Chandra and the flowering of abstract harmonic analysis.

(3) The whole industry of root systems evinced in the writings of I. Macdonald, V. Kac, R. Moody, and others started with Killing. For the latest see [21].

(4) The Weyl group and the Coxeter transformation are in Z.v.G.II. There they are realized not as orthogonal motions of Euclidean space but as permutations of the roots. In my view, this is the proper way to think of them for general Kac-Moody algebras. Further, the conditions for symmetrisability which play a key role in Kac's book [17] are given on p. 21 of Z.v.G.II.

(5) It was Killing who discovered the exceptional Lie algebra $E_8$, which apparently is the main hope for saving Super-String Theory -- not that I expect it to be saved!

(6) Roughly one third of the extraordinary work of Elie Cartan was based more or less directly on Z.v.G.II.

Does English translation of Z.v.G.II exist?

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  • $\begingroup$ Well, Coleman in his paper exactly complains about that nobody has ever read Killing‘s work. I guess one reason for this is that there was no translation. $\endgroup$
    – ThiKu
    Commented Oct 4, 2018 at 5:57
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    $\begingroup$ there was a time when German was a language of scientific communication, which is presumably why this was never translated; perhaps one could crowd-fund a translation, I would estimate the time for a translation of the text without retyping the formulas at 20 minutes per page, so it should be about 16 hours of work. $\endgroup$
    – Carlo Beenakker
    Commented Oct 4, 2018 at 19:26
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    $\begingroup$ If it is so important paper, maybe it will be worthwhile to translate it and put the translation at least on arXiv. Maybe German mathematicians can do it? $\endgroup$
    – Zurab Silagadze
    Commented Oct 5, 2018 at 8:26
  • $\begingroup$ It is such an important paper because it was the basis for the work of Élie Cartan (and then of Hermann Weyl and his followers), i.e., for all the stuff that you learn today in a course on semisimple Lie algebras. If you want to learn this material by now, you are probably better off with reading one of the existing textbooks. $\endgroup$
    – ThiKu
    Commented Oct 5, 2018 at 8:39
  • $\begingroup$ One difficulty is that the print is fairly low resolution so that the indices require some guessing. So maybe a first step would be to transform the paper into digital form. $\endgroup$
    – Manfred Weis
    Commented Oct 5, 2018 at 19:20

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This paper has been translated by David Delphenich under the title of "The composition of continuous, finite, transformation groups" (see here).

[Note that the paper's date is given as 1869 in the translation's heading, but the paper itself refers to Killing's publications and correspondence in the 1880s.]

Edit: apologies to all, I should have read the paper's title more carefully. The translation covers only the first part, not the second part as requested. But the translator does accept requests.

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    $\begingroup$ That's a serious typo (1869 for 1889), maybe better inform the translater? $\endgroup$
    – YCor
    Commented Feb 11 at 10:01

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