Francesco Cattafi

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Harish-Chandra's proof of Lie's third theorem

Hi,

About your recent addition of remarks on proofs of Lie's third theorem (which are very nice). If I remember correct, I heard that there was a proof of the theorem by Harish-Chandra. Do you know anything about it? Is it one of the three mentioned in the article? (I did a quick Google search but couldn't find anything so maybe my memory was wrong.) Taku (talk) 08:16, 19 September 2022 (UTC)

Hi, thanks for the question! Unfortunately I have never heard about such proof, but after a bit of searching online I found these notes, https://math.mit.edu/~ngtriant/notes/ados_theorem.pdf, where they mention that Harish-Chandra actually provided in 1949 a proof not of Lie's third theorem but of Ado's theorem. Indeed, I just read now that in the page Ado's theorem they mention his paper among the references. Unfortunately they don't mention anything else about the actual proof(s) of the result, and I have actually never read it myself so I wouldn't be able to improve that article (but it would be nice if someone else could!) Francesco Cattafi (talk) 16:36, 25 September 2022 (UTC)

First my apology for the late response. Second, thank you for doing some digging in history. After reading your response, I do now think I misremember; it was Ado's history not Lie's third theorem (though, of course, they are closely related). In any case, the right place to mention Harish-Chandra's proof is therefore at the Ado's theorem article. If I can find time, I will try to read Harish-Chandra's proof and see if it differs from the standard one (and edit the article accordingly). Anyway thank you again for the response. -- Taku (talk) 09:00, 18 November 2022 (UTC)

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