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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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  • $\begingroup$ 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. $\endgroup$ Commented Aug 2, 2011 at 16:35
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    $\begingroup$ 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). $\endgroup$ Commented Aug 2, 2011 at 17:11
  • $\begingroup$ @Qiaochu: maybe you're right. Which book can you recommend on Lie theory? $\endgroup$ Commented Aug 2, 2011 at 18:30
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    $\begingroup$ I found Howe's short article Very basic Lie theory brilliant. $\endgroup$ Commented Aug 2, 2011 at 20:06
  • $\begingroup$ Does this answer your question? What is a good book for a second "course" in group theory? $\endgroup$ Commented May 30, 2023 at 5:39

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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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  • $\begingroup$ Publisher's page here. $\endgroup$ Commented Aug 2, 2011 at 19:20
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    $\begingroup$ Permanent link here. $\endgroup$ Commented Aug 2, 2011 at 20:07
  • $\begingroup$ 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? $\endgroup$ Commented Aug 3, 2011 at 0:50
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    $\begingroup$ @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. $\endgroup$ Commented Aug 30, 2011 at 12:14
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    $\begingroup$ @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. $\endgroup$ Commented Aug 31, 2011 at 5:02
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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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