## Learning about Lie groups

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Can someone suggest a good book for teaching myself about Lie groups? I study algebraic geometry and commutative algebra, and I like lots of examples. Thanks.

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I haven't seen Taylor's "Several Complex Variables" mentioned, but coming from an algebraic geometry background myself, I found that book very nice. It does use a lot of analysis though (a lot for me, anyway). – Steve D Jul 21 2010 at 16:33

## 16 Answers

There's also Fulton & Harris "Representation Theory" (a Springer GTM), which largely focusses on the representation theory of Lie algebras. Everything is developed via examples, so it works carefully through $sl_2$, $sl_3$ and $sl_4$ before tackling $sl_n$. By the time you get to the end, you've covered a lot, but might want to look elsewhere to see the "uniform statements". An excellent book.

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@Scott I agree and it's one of the few introductory books that develops why Lie groups are important as algebraic objects in thier own right rather then just tools in differential geometry. – Andrew L Jun 24 2010 at 19:28

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I realize this answer is rather late, but I just wanted to mention a fairly recent book on Lie theory that offers a gentle introduction to the basics: John Stillwell's Naive Lie Theory. It does not cover representation theory, but might be a pleasant step up to a book that does. The level is advanced undergraduate.

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For someone with algebraic geometry background, I would heartily recommend Procesi's Lie groups: An approach through invariants and representations. It is masterfully written, with a lot of explicit results, and covers a lot more ground than Fulton and Harris. If you like "theory through exercises" approach then Vinberg and Onishchik, Lie groups and algebraic groups is very good (the Russian title included the word "seminar" that disappeared in translation). However, if you want to learn about the "real" side of Lie groups, both in linear and abstract manifold setting, my favorite is Godement's "Introduction à la théorie des groupes de Lie".

Several of the books mentioned in other answers are devoted mostly or entirely to Lie algebras and their representations, rather than Lie groups. Here are more comments on the Lie group books that I am familiar with. If you aren't put off by a bit archaic notation and language, vol 1 of Chevalley's Lie groups is still good. I've taught a course using the 1st edition of Rossmann's book, and while I like his explicit approach, it was a real nightmare to use due to an unconscionable number of errors. In stark contrast with Complex semisimple Lie algebras by Serre, his Lie groups, just like Bourbaki's, is ultra dry. Knapp's Lie groups: beyond the introduction contains a wealth of material about semisimple groups, but it's definitely not a first course ("The main prerequisite is some degree of familiarity with elementary Lie theory", xvii), and unlike Procesi or Chevalley, the writing style is not crisp. An earlier and more focused book with similar goals is Goto and Grosshans, Semisimple Lie algebras (don't be fooled by the title, there are groups in there!).

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Brian Hall's "Lie Groups, Lie Algebras and Representations: An Elementary Introduction" specializes to matrix Lie groups, so it makes for an accessible introduction. Like Fulton & Harris, it's got plenty of worked examples. It also has some stuff about Verma modules that's not in Fulton & Harris. I think it'd be a great book for a first course.

Knapp's "Lie Groups: Beyond an Introduction" might be good for a second course (it has more of the "uniform statements" Scott mentioned) and is handy to have around as a reference. It has an appendix with historical notes and a ton of suggestions for further reading. It also has a lot more on Lie groups themselves than most books do.

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Knapp's book is a lot harder then either F&H or Hall. Hall's really more for physics majors,but it's a nice book nevertheless. – Andrew L Jun 24 2010 at 19:29

I like Humphreys' book, Introduction to Lie Algebras and Representation Theory, which is short and sweet, but doesn't really talk about Lie groups (just Lie algebras). I also sometimes find myself looking through Knapp's Lie Groups: Beyond an Introduction. If the material was covered in the Spring 2006 Lie groups course at Berkeley, then I prefer the presentation in this guy's notes.

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Before the 2006 course, there was Allen Knutson's 2001 course, from which there are several sets of notes, e.g. math.berkeley.edu/~allenk/courses/spr02/261b/… – Scott Morrison♦ Sep 28 2009 at 23:36

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Also, Theo Johnson-Freyd has some notes from Mark Haiman's Fall 2008 course here: math.berkeley.edu/~theojf/LieGroupsBook.pdf – Anton Geraschenko♦ Sep 29 2009 at 0:00

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Finding a reasonably elementary book on Lie groups with lots of examples is challenging. What makes the subject attractive is that it's the crossroads for many subjects. My book definitely wasn't about Lie groups (and has too few examples) but does get somewhat into "modern" representation theory. Knapp is reliable but somewhat advanced. Fulton-Harris is also not a Lie group book and doesn't introduce infinite dimensional representations, but covers a lot of concrete classical examples plus symmetric groups. Free online notes can be a safe starting point, but shop around. – Jim Humphreys Jun 24 2010 at 14:04
@Anton You and Theo should be both be very proud of your TeX-ed notes,particularly on Lie theory. – Andrew L Jun 24 2010 at 19:37

Dan, knowing your tastes, I think you will like Fulton-Harris very much. However, if I recall correctly, Fulton-Harris doesn't go into much depth about some important (and really cool) theorems in Lie groups, such as Peter-Weyl and Borel-Weil-Bott. But of course, you can learn these theorems elsewhere.

I think the book "Compact Lie Groups" by Sepanski is nice, and it does cover P-W and B-W-B. I also found this note on B-W-B to be useful in the past: http://www-math.mit.edu/~lurie/papers/bwb.pdf

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Nice link on BWB. I had not seen that before. – B. Bischof Dec 18 2009 at 15:34

Just to add one more to the already mentioned. I find the book of Bump on Lie groups very good, as well as the other ones.

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The book "Introduction to Lie groups and Lie algebras" by A. Kirillov, Jr., is quite nice, and seems to be free online. It might be a good starting point, and it has an excellent annotated bibliography. (Edit: On further inspection, the .pdf I linked to just seems to be a draft. The actual book has the good bibliography.)

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I haven't carefully read that one,but a lot of my friends who work in Lie theory swear by it.  – Andrew L Jun 24 2010 at 19:30

Although perhaps not from the point of view of someone interested in algebraic geometry and commutative algebra, others of different persuasions might enjoy the following books:

• Lectures on Lie groups, by J. Frank Adams
• Representations of compact Lie groups, by Theodor Bröcker and Tammo tom Dieck
• Lie groups: an introduction through linear groups, by Wulf Rossmann

Adam's book is a classic and has a very nice proof of the conjugacy theorem of maximal tori using algebraic topology (via a fixed point theorem). Bröcker-tom Dieck is a good companion to Adams, as it often reads like an expanded version of it. At any rate, it goes into more detail. Rossmann's book is reviewed by Knapp in http://www.math.sunysb.edu/~aknapp/pdf-files/BakerRossmann.pdf

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Adams's book is my favorite - it is a real gem. – Jeffrey Giansiracusa Jun 24 2010 at 10:14

I really like Goodman & Wallach. This is a new revised version of their old book which was called, "Representations and Invariants of the Classical Groups". It is really clearly written and covers a lot of material. It might suit your interests, since it's a bit bent towards the algebraic groups part of Lie theory, but it does also cover the analytic side.

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Roger Godement-Introduction a la theorie des groupes de Lie-Springer(only in french as far as I know).An introduction to Lie groups via linear groups(with John von Neumann in backstage..) and a touch of Hilbert 5th problem...Very fun, as always with Godement

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My favourite reference is Serre, Lie algebras and Lie groups. It's a tour of Bourbaki's Lie groups and Lie algebras that is concise and, being Serre, of course, very clear.

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As an elementary introduction with lots of examples you may take a look at A.Baker,"Matrix Groups. An Introduction to Lie Group Theory" which appeared in Springer's Undergraduate Texts in Mathematics. After this a very good book with lot of results and almost self-contained, but rather demanding is M.M.Postnikov "Lie Groups and Lie Algebras" (it was published by "Mir" in English).

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There are many courses, including something about Lie groups at J.Milne's page: jmilne.org

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Nobody mentioned "Gilmore: Lie Groups, Physics, and Geometry" yet.

A very down to earth introduction with many examples and clear explanations. Especially targeted at physicists, engineers and chemists.

If you follow the above link you can read some sample chapters.

The cover summarizes the set up of the book quite neatly:

"Describing many of the most important aspects of Lie group theory, this book presents the subject in a ‘hands on’ way. Rather than concentrating on theorems and proofs, the book shows the relation of Lie groups with many branches of mathematics and physics, and illustrates these with concrete computations. Many examples of Lie groups and Lie algebras are given throughout the text, with applications of the material to physical sciences and applied mathematics. The relation between Lie group theory and algorithms for solving ordinary differential equations is presented and shown to be analogous to the relation between Galois groups and algorithms for solving polynomial equations. Other chapters are devoted to differential geometry, relativity, electrodynamics, and the hydrogen atom. Problems are given at the end of each chapter so readers can monitor their understanding of the materials. This is a fascinating introduction to Lie groups for graduate and undergraduate students in physics, mathematics and electrical engineering, as well as researchers in these fields.

Robert Gilmore is a Professor in the Department of Physics at Drexel University, Philadelphia. He is a Fellow of the American Physical Society, and a Member of the Standing Committee for the International Colloquium on Group Theoretical Methods in Physics. His research areas include group theory, catastrophe theory, atomic and nuclear physics, singularity theory, and chaos."

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In my opinion, the best quick introduction to Lie group and algebra theory is in chapter 12 of E. B. Vinberg's A Course In Algebra. It is short, geometric and deep with all the essential facts and theorems presented. There's a similar presentation in Artin's Algebra, but that one is done entirely in terms of matrix groups. The Vinberg chapter is on general Lie theory. By the way, it's mostly drawn from the Vinberg/Onischick book mentioned by Victor above -- but it's a little gentler and more detailed, being pitched at beginners.

The Vinberg book is one of those texts you read over and over because every time you look at it, you realize a little more just how damn good it is.

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