Could you recommend some books on Lie algebra？ I am a pure maths student, and  want to go straight ahead, so I  decide to study Lie algebra on my own, and try my best to understand it from various points of view：differential equation, Lie group, representation theory,..
Could you provide some advice and recommend some books? Thanks a lot
 A: This. Easy to read. You can read it like a Harry Potter storybook.
http://www.amazon.com/Introduction-Algebras-Springer-Undergraduate-Mathematics/dp/1846280400
A: Naive Lie Theory by Stillwell. 
A: These free notes by Alistair Savage are an excellent introduction (based on Stillwell's and Hall's books). A bit more advanced, yet inclusive of Stillwell.
http://alistairsavage.ca/mat4144/notes/MAT4144-5158-LieGroups.pdf
A: The reason why you want to study Lie algebras can have a great impact on what books one would recommend. Do you want to study solely the algebraic side? With a view towards algebraic groups? As a second introduction to representation theory (after finite groups)? Or do you want to learn about Lie theory, i.e. Lie groups and Lie algebras?

Without more information, I would explain what I did when I wanted to learn about Lie algebras. For background, I'll just say that I was interested in algebraic groups, and later got interested in number theory and automorphic forms (and so I then had to go back and learn about Lie groups). I started with Introduction to Lie algebras by Erdmann and Wildon. This is very hands down, they assume right away that you are working over the complex numbers. You won't get quite far with this book (it covers the main definitions and gives the structure theorem for semisimple Lie algebras), but if you do the exercises, you will have a good foundation. Then I moved to Humphreys' Introduction to Lie Algebras and Representation Theory (which has already been mentioned and is the absolute best). It is more terse than Erdmann and Wildon, and the exercises are more difficult, but it covers more.
Then, you might want more heavy-duty stuff. That's when I went to Lie Groups Beyond an Introduction by Anthony Knapp. For this, you need some knowledge of topology and differential geometry, i.e. knowledge of smooth manifolds. But this is a very good book, and it covers a wide range of topics.
A: James Humphreys - Introduction to Lie Algebras and Representation Theory (link)
