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I have been using the concepts of Lie group, Lie algebra and some of its properties for quite a while now in various geometry courses, but I had to pick those concepts along the way because they are always taken for granted. I want to fill this hole and I was wondering which (graduate) book will be best suited for this, some research led me to this list, although I am open for any suggestion:

-Brian C. Hall, Lie groups, Lie algebras and representations. Currently my least favorite option, mainly because of the answer given here. I do know differential geometry and I would like to study this subject in all generality. Still people seem to like this book, and it has a lot of problems, which I appreciate very much.

-V. S. Varadarajan, Lie groups, Lie algebras and their representations. I know very little about this book, I have been told that it is kind of a more general version of the last one (using differential geometry freely), but I did not manage to find a copy on any library to check it out. Does it have a good amount of problems?

-J. Harris and W. Fulton, Representation theory. Seems to be a classic, but (and I am judging just by the title for I could not get a copy of this one either) I am afraid this book will go too fast on the basics of Lie groups and Lie algebras. Also, how about the problems on this one?

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Hall's book is excellent. You can't go wrong there.

I would also suggest supplementing with Chapter 4 of Tu's book for more of a complete connection with the geometry (Hall's book largely focuses on the representation theory of Lie Groups and Lie Algebras, although there is geometry in that too in later chapters).

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  • $\begingroup$ My main concern with this one is that it unnecessarily avoids talking about geometry from the beginning, so I am worried that would translate into a less thorough treatment of the subject. If you had a chance to look at Fulton's, how would you compare them? $\endgroup$ – Smurf Jan 18 '18 at 9:35
  • $\begingroup$ I don't think I'm a great source for opinions on Fulton and Harris - I've used parts of it at times, but never thoroughly read it or worked through the problems. I will say though that my impression is that it's a decent reference book, but not a good introduction (ironic given the title). The reason I say this is that the proofs are very brief and often alluded to or described in words, rather than in maths, which is a personal pet peeve. At least when I've needed to use it. The problems are probably decent as per your original question, but that's true of any book on the topic these days imo $\endgroup$ – Jonathan Rayner Jan 18 '18 at 11:28
  • $\begingroup$ Further, I see no reason to read Fulton over Hall if you're specifically interested in Lie Groups. As I've said and have been mentioned in other answers on this site, the only possible draw back of Hall is you might want to focus on geometry over representation theory. Fulton's book is a book on representation theory, so that gets you nowhere if you want more geometry :) $\endgroup$ – Jonathan Rayner Jan 18 '18 at 11:43
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Try Chevalley,Claude Theory of Lie Groups (Dover) .An oldy but a goody .This book is a classic ,self contained for the most part and logically very clear .

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    $\begingroup$ Some remarks on how this "oldy" relates or compares to the books already provided in the body of the Question, or otherwise proffering how the book by Chevalley meets the request, would elevate this post from Comment-quality content to an Answer. $\endgroup$ – hardmath Jan 14 '18 at 6:34
  • $\begingroup$ Managed to take a look of this one. It seems to be missing a big amount of material with respect the other books I am considering and, most importantly, it has no problems. $\endgroup$ – Smurf Jan 18 '18 at 9:33

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