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I am currently learning about Lie groups,So kindly suggest a reference for Lie groups, which contains lecture on Manifolds as well.

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    $\begingroup$ I found Warner's book "Foundations of Differentiable Manifolds and Lie Groups" to be good. Although I haven't looked at the chapters on basic manifold theory. Lee's "Introduction to Smooth Manifolds" also has some basic Lie groups stuff. His books are very well written in my opinion. $\endgroup$ Commented Apr 6, 2014 at 7:04
  • $\begingroup$ I would recommend the Naive Lie Theory by John Stillwell springer.com/gp/book/9780387782140 $\endgroup$
    – IgotiT
    Commented May 5, 2021 at 16:01

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The best book I found on the basics of Lie theory is Structure and Geometry of Lie Groups by Hilgert and Neeb, which for some reason is not very well known. It contains a chapter on manifolds, but if you never studied them before it might be a bit rushed. It barely has anything on semisimple Lie groups, though (only the Cartan and Iwasawa decompositions, without the classification of all semisimple Lie algebras and groups using root systems).

I agree it's probably better to study manifold theory first, and I definitely recommend John Lie's book Introduction to Smooth Manifolds (although it contains much more than you need).

After studying manifold theory separately you can get back to Hilgert & Neeb's book. Also there is a book by Kirillov and a book by Varadarjan (which actually also contains a chapter on manifolds, but it is definitely too brief if you haven't seen it before); both books are nice. They're less friendly than Hilgert & Neeb, but they go deeper into the theory of Semisimple Lie groups (or at least semisimple Lie algebras in Kirillov's case).

If like me you still don't feel comfortable with manifolds even after you learned about them separately, though, I don't recommend Varadarajan's book. He goes deeper into their theory than Hilgert & Neeb (and Kirillov).

There is also the book by Knapp. It is nice, but only if you already know the elementary theory of Lie groups (say by reading chapter 9 of Hilgert and Neeb).

Two other common recommendations are Warner's book and the book by Onishchik & Vinberg. Warner book is written pretty nicely, but he goes way deeper into differential geometry than you need, and it barely contains anything on Lie groups other than the extremely basic stuff (although it probably contains enough to start reading Knapp's book).

Onishchik & Vinberg's book is (in my opinion) very hard. In order to read it you need to be prepared to read at a very slow pace and solve all the problems (otherwise you won't really understand it). Also, you will have to study the theory of manifolds separately for the first chapter, and it is probably a good idea to study algebraic geometry and algebraic groups separately in order to understand chapters 2-3, which are necessary for chapters 4-5. If you're willing to do all that (and have like a year to spare for it), than it will be great for you. He covers the basics of Lie groups, algebraic groups, and semisimple (complex or real) Lie groups.

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I think if a book's goal is to cover Lie groups, then a prologue on manifolds often feels rushed and you get a watered-down version. You should take a good book on Lie groups and hold it next to a good book on manifolds, and switch between them when you need to look something up.

For manifolds I can easily say Smooth Manifolds by Lee is great. For Lie groups I think it's a bit harder ... Good books are by: B. Hall; Brocker and van Dieck; and Fulton and Harris. (Just look up the author's name with keyword "Lie groups".) Fulton/Harris does a lot on Lie algebras that you might not be interested in, but it's useful to have it as a reference to get a full picture of the theory.

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