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Currently I'm studying differential geometry and PDEs - so I often meet the use of groups. I also studied symmetries methods for solutions of differential equations but the connection between Lie groups and Lie algebras is still not implicit for me.

I am looking for a literature which gives a nice description of group theory (I am especially interested in continuous groups) and necessary covers Lie groups and Lie algebras. Thanks in advance.

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I don't much about this subject: But a friend a mine recently attended a $\text{Lie Algebra}$ workshop and was told to refer: $\text{Introduction to Lie algebras and representation theory}$ by Humphreys. –  user9413 Aug 2 '11 at 16:35
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When most people say "group theory" they generally mean the study of abstract groups, not of topological etc. groups. It sounds like you really want a book on Lie theory (that maybe covers applications to differential geometry and PDE). –  Qiaochu Yuan Aug 2 '11 at 17:11
    
@Qiaochu: maybe you're right. Which book can you recommend on Lie theory? –  Ilya Aug 2 '11 at 18:30
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I found Howe's short article Very basic Lie theory brilliant. –  t.b. Aug 2 '11 at 20:06

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I think that Naive Lie Theory is an excellent place to start. Once you find what you want to know in it, it has excellent references for where to continue.

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Publisher's page here. –  lhf Aug 2 '11 at 19:20
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Permanent link here. –  t.b. Aug 2 '11 at 20:07
    
I remember reading it and not learning anything about the representation theory of Lie algebras, which is a deficiency in my mind given its use in modern theoretical physics. (And I still don't know anything about it...) Could you suggest an introductory text which covers representations? –  Zhen Lin Aug 3 '11 at 0:50
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@D B Lim: I know about linear representation of finite groups. The representation theory of Lie groups and Lie algebras is a different but related beast. –  Zhen Lin Aug 30 '11 at 12:14
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@Zhen I thought An Introduction to Lie Algebras and Representation Theory by James Humphreys was an excellent (Atiyah and Macdonald "style") textbook on Lie algebras. –  Amitesh Datta Aug 31 '11 at 5:02

For your interests, I would recommend Sattinger and Weaver's Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics. They introduce the basics of Lie theory, and the applications they have in mind are all related to differential equations.

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