Reference request: group theory

Question

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.

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. — , Aug 02 '11 at 16:35
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 02 '11 at 17:11
@Qiaochu: maybe you're right. Which book can you recommend on Lie theory? — SBF, Aug 02 '11 at 18:30
I found Howe's short article Very basic Lie theory brilliant. — t.b., Aug 02 '11 at 20:06
Does this answer your question? What is a good book for a second "course" in group theory? — , May 30 '23 at 05:39
@user32570 thanks. I’m not sure whether I’ll be reading it anytime soon though — SBF, May 30 '23 at 06:03
@user32570 This is not a duplicate. The answers to both questions are very different (and none of the books suggested there cover Lie algebras or Lie groups). — user1729, May 31 '23 at 07:56

score 5 · Accepted Answer · edited Aug 02 '11 at 19:19

5

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.

edited Aug 02 '11 at 19:19

lhf

221,500

answered Aug 02 '11 at 18:54

davidlowryduda

94,345

Publisher's page here. – lhf Aug 02 '11 at 19:20
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Permanent link here. – t.b. Aug 02 '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 03 '11 at 00:50
@Zhen Lin If you're interested in representations you could go for Serre's linear representation of finite groups.....but if you would like an introductory text chapter 10 of Michael Artin's Algebra may be helpful – Aug 30 '11 at 11:42
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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 05:02

score 1 · Answer 2 · answered Jan 28 '14 at 06:59

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.

Reference request: group theory

2 Answers2