I have a background in representation theory of the symmetric group and GL(n), and I need to learn quickly about representations of (semisimple) Lie algebras in order to follow a course I'm interested in. The books I'm using assume you don't know anything about representations. However, I feel I could be learning quicker if there was a book which more closely related both subjects, or maybe highlighted similarities and differences between the two theories. What sources would you recommend?
Learning representation of Lie algebras for someone who knows representations of the symmetric group
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4$\begingroup$ The quickest way to get the basics is in my opinion with J.E. Humphreys's Introduction to Lie Algebras and Representation Theory (GTM 9). The big advantage of this book in contrast to others is, that you can look up things without having to read an entire chapter. It is also very good to read and the chapters aren't too long. $\endgroup$– Marius S.L.Oct 24, 2021 at 18:37
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2$\begingroup$ Representations of Lie algebras are much more linear algebra than group representations, because here both the Lie algebra and the representation space are vector spaces. So just review these parts of linear algebra, and a bit of abstract algebra for ideals etc. Also review bilinear forms and look up what the Killing form is, and weights and roots. This will give you want you want - "need to learn quickly about representations of (semisimple) Lie algebras." $\endgroup$– Dietrich BurdeOct 24, 2021 at 18:54
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My suggestions are
B. Hall; Lie Groups, Lie Algebras, and Representations. It gives you a complete and simple introduction to Lie groups, Lie algebras etc. I would take this approach even if you do not immediately need the "Lie group" part and only the "Lie algebra" part since if you don't it's easy to get confused later on when you miss a lot of the motivations and the implications of some constructions.
K. Erdmann; Introduction to Lie Algebras. If you just want the "Lie algebra" part. You can't get a simpler introduction to Lie Algebra. There are also some YouTube lessons that follow step by step the book.
