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I'm a physics student trying to learn something about the classification of compact simple Lie algebras. I asked one of my TA's for a book recommendation and got Fulton's "Representation Theory". After giving it a shot and reading chapters 7 and 8 (with which's topics I'm already familiar with) I can't really say that I'm understanding a lot. It's hard to say why, but I guess I'm not a huge fan of the way how the author explains (words) things, and I'm missing the mathematical structure (Lemma, Prop., Thm, Proof, etc.).

Googling the topic gives a lot of results, but many of these are semester projects or thesis from other people and not really textbooks. I don't mind reading those, but I'm not really sure if these are the most pedagogical approach to learn something about the topic. Some of the above mentioned semester projects/thesis:

The last one actually looked quite promising, but I haven't really had time to check it in a bit more detail. My question therefore boils down to: What textbooks/references can you recommend for the topic "Classification of compact simple Lie algebras"?

PS:

  • I've already had a proper two semester course on representation theory, starting with groups and their properties in a abstract form, then to represenations in general, then to Lie algebra representations, etc. So I have some basic understanding about these things. I saw that in the context of representation theory some concepts of differential geometry are mentioned (manifolds, differentials, etc.), you can also assume that a basic knowledge is present in this area as well.
  • I've seen this question, which kind of relates to mine, but it seems to focus on much narrower questions than mine, so I don't really think it qualifies as duplicate.
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  • $\begingroup$ That is a difficult question. Usually people interested in Lie algebras are either physicists who want something in the style of the Georgi classic (Lie Algebras in Particle Physics) or pure mathematicians who want the Lie Group material covered in depth. I agree with you that the Fulton book is an odd choice for Lie Algebras (even though I like it for other reasons). $\endgroup$
    – almagest
    Jan 30, 2020 at 14:43
  • $\begingroup$ On top of Dietrich Burde's excellent pointers here, compare the answers to math.stackexchange.com/q/3121110/96384. $\endgroup$
    – Torsten Schoeneberg
    Jan 30, 2020 at 17:35

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For the classification of finite-dimensional complex simple Lie algebras I can indeed recommend Humphrey's book, Knapp's book, and in particular the diploma thesis by Florian Wisser. Fulton's book classifes also all finite-dimensional representations of these Lie algebras. This is more than what you ask for in your title. Perhaps this was a misunderstanding. In physics you definitely want both.

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  • $\begingroup$ I don't need the represantations, at least not for now. Thank you very much for the recommendations, I'll make sure to check them all out! $\endgroup$
    – Sito
    Jan 30, 2020 at 15:33
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    $\begingroup$ OK, good. Knapp's book is called "Lie Groups Beyond an Introduction", but also has a chapter on the classification of simple Lie algebras. $\endgroup$
    – Dietrich Burde
    Jan 30, 2020 at 15:42

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