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Learning about Lie groups

Actually, I'm having the hard time with Serre's book on Lie groups and algebras: the lack of motivation is my biggest problem. So, what would you suggest for a first, illustrative, but systematic and deep course on Lie groups? The lack of good books on Galois theory on my way is a different problem: too much formalism without much of results is what I've seen in Postnikov's "Galois theory" so far

As a little note on preferences: right now I'm enjoying the topology book of Seifert and Threlfall(with its geometric illustrations) and homological algebra by Cartan & Eilenberg(because of the well-understandable language of diagrams).

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marked as duplicate by Victor Protsak, S. Carnahan Aug 17 '10 at 15:41

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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First and deep are, I think, bad things to require simultaneously; you would probably get better answers if you picked one or the other. –  Qiaochu Yuan Aug 17 '10 at 7:27
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Part of the trouble here is that learning examples is very important to the study of Lie Theory. I think most students do better when they first become familiar with the classical groups, their representations and homogeneous spaces, and then move to a systematic study. If you aim to follow this route, I would start with Fulton and Harris, then try Knapp. –  David Speyer Aug 17 '10 at 12:01
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Is this really an MO question? (If so, it invites a very long list and should be community wiki.) There's more than one subject involved here as well. Anyway, answers always depend on too many variables: background and potential interests of the person asking, plus resources available including library, internet; the tastes of all those others who have learned subjects in various ways. Some people love one source, others hate it. –  Jim Humphreys Aug 17 '10 at 13:20
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I don't see any reason for combining two topics in one question. For Lie groups, this is a duplicate of mathoverflow.net/questions/13/learning-about-lie-groups –  Victor Protsak Aug 17 '10 at 14:24

6 Answers 6

Concerning Lie groups and Lie algebra, I suggest Knapp's "Lie groups, beyond an introduction". It starts with a chapter 0 on classical matrix groups, then goes on to the general theory.

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Frank Warner's book "Foundations of differentiable manifolds and Lie groups" is one of the standards. You can't go wrong by looking at Chevalley's book "Theory of Lie groups" or Weyl's (classic, of course) "The classical groups: their invariants and representations". Knapp's big book "Lie groups: beyond an introduction" has lots (and lots) of information.

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It's worth telling a beginner not to confuse the general foundations and Lie's three theorems (the focus in the books of Serre and Warner) with the deeper core of Lie theory, which is the structure theory of semisimple Lie groups and Lie algebras and their representations, aspects of which are addressed to various degrees in the books of Chevalley and Weyl and Knapp (and Hochschild's "Structure Theory of Lie groups", which includes the complex-analytic case but whose style makes it perhaps best appreciated after one has some experience). Also look at Serre's thin book on complex ss Lie alg. –  BCnrd Aug 17 '10 at 12:24

Patrick Morandi's Field and Galois Theory is a good book for beginners. He gives lots of examples and has interesting exercises too. For a later reading though, I would suggest the Galois theory section in Lang's Algebra.

I really liked Hsiang's Lectures in Lie Groups although it may be a bit short for a full course. And Kirillov Jr.'s book Introduction to Lie Groups and Lie Algebras (also available as a published book) is a very good introduction to the topic with plenty of nice examples in the exercises. And lastly, Serre's Complex Semisimple Lie Algebras is great once you manage to get through it, i.e., it's a gem but not for the first reading!

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1.Lie Groups: M.Postnikov Lie Groups and Lie Algebras-vol5 of his Lectures in Geometry; as a bonus not systematic but deep and witty: Roger Godement-Introduction à la théorie des groupes de Lie(french)

2.Galois Theory: H.M Edwards -Galois Theory Not much "abstract nonsense", with historical insight

V.B. Alekseev-Abel's Theorem In Problems And Solutions: Based on the lectures of Professor V.I.Arnold

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Ian Stewart's Galois Theory is a nice introductory text to Galois theory. Recently, however, I've been doing exercises from chapters 13 and 14 from Abstract Algebra by Dummit and Foote. This might be a faster introduction. Joseph Rotman's short book Galois Theory is also introductory, but fast and very readable.

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As always it depends on what you know (i.e. your background) and on what you need. For Galois theory, there is a nice book by Douady and Douady, which looks at it comparing Galois theory with covering space theory etc. Another which has stood the test of time is Ian Stewart's book.

For Lie groups and Lie algebras, it can help to see their applications early on, so some of the text books for physicists can be fun to read. They skip some detail but provide the intuition that is sometimes lacking in purely mathematical texts.

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