What’s a good place to learn Lie groups?

Ok so I read the following article the other day: http://www.aimath.org/E8/ and I wanted to learn more about lie groups. Using my exceptional deduction skills I thought “oh it must have something to do with groups” So I picked up a copy of Dummit and Foote’s book on abstract algebra and skimmed through it. It didn’t say anything about Lie groups however. $E_8$ is coming to be rather famous so maybe other people are interested in this question too. Lets suppose I wanted to learn about lie groups. What books should I read to be ready to learn about Lie groups and what is a good book that talks about Lie groups. I’m guessing its a combination of group theory (representation theory in specific) and also differential geometry. Is this correct? Thank you very much for your time.

Answer

I think a good place to start with Lie groups (if you don’t know Differential Geometry like me) is Brian Hall’s Book Lie Groups, Lie algebras and Representations. The strength of such a book for me would be that it talks about matrix Lie groups, e.g. $SO(n),U(n),GL_n, Sp_n,SL_n$ and not general Lie groups in terms of abstract manifolds. Furthermore, the Lie algebra is introduced not as an abstract linear space with a bracket but as the set of all matrices $X$ such that $e^{tX}$ lands in the matrix Lie group for all $t$.

I am using this book now for a course and I find it extremely readable. For one, proofs are presented in almost complete detail and it is easy to follow. By this I mean one does not need a lot of prerequisites to understand the material. You should of course have an understanding of linear algebra, as well as know topological concepts like connectedness, compactness and path-connectedness.

In conclusion, I think the main strength of Hall’s Book is that it teaches you ideas through lots and lots of examples. For example, an entire chapter (IIRC chapter 5) is devoted entirely to the representation theory of the Lie algebra $\mathfrak{sl}_3(\Bbb{C})$. I learned a lot from that example there!

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Source : Link , Question Author : Asinomás , Answer Author : Community

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