# What's a good place to learn Lie groups?

Ok so I read the following article the other day: http://www.aimath.org/E8/ and I wanted to learn more about lie groups. Using my exceptional deduction skills I thought "oh it must have something to do with groups" So I picked up a copy of Dummit and Foote's book on abstract algebra and skimmed through it. It didn't say anything about Lie groups however. $E_8$ is coming to be rather famous so maybe other people are interested in this question too. Lets suppose I wanted to learn about lie groups. What books should I read to be ready to learn about Lie groups and what is a good book that talks about Lie groups. I'm guessing its a combination of group theory (representation theory in specific) and also differential geometry. Is this correct? Thank you very much for your time.

I think a good place to start with Lie groups (if you don't know Differential Geometry like me) is Brian Hall's Book Lie Groups, Lie algebras and Representations. The strength of such a book for me would be that it talks about matrix Lie groups, e.g. $SO(n),U(n),GL_n, Sp_n,SL_n$ and not general Lie groups in terms of abstract manifolds. Furthermore, the Lie algebra is introduced not as an abstract linear space with a bracket but as the set of all matrices $X$ such that $e^{tX}$ lands in the matrix Lie group for all $t$.

I am using this book now for a course and I find it extremely readable. For one, proofs are presented in almost complete detail and it is easy to follow. By this I mean one does not need a lot of prerequisites to understand the material. You should of course have an understanding of linear algebra, as well as know topological concepts like connectedness, compactness and path-connectedness.

In conclusion, I think the main strength of Hall's Book is that it teaches you ideas through lots and lots of examples. For example, an entire chapter (IIRC chapter 5) is devoted entirely to the representation theory of the Lie algebra $\mathfrak{sl}_3(\Bbb{C})$. I learned a lot from that example there!

You don't need to know any differential geometry to grasp the basic ideas in Lie theory beyond some idea of what a tangent vector is. The study of semisimple Lie groups (which includes $E_8$) is largely algebraic (there are theorems that make this precise but you don't need to know what they are) and getting a good grasp of the important examples doesn't require more than comfort with calculus and linear algebra.

I would recommend Stillwell's Naive Lie Theory in this vein. I agree with Matt E that Fulton and Harris is also a solid resource.

• For a more specific recommendation of what you should aim for, aim to have a good understanding of what $\text{SU}(2)$ and $\text{SO}(3)$ look like, their representations, and the relationship between them. This is already extremely useful (you can now study spin and angular momentum in quantum mechanics) and it is also important for understanding more complicated Lie groups. – Qiaochu Yuan Sep 12 '12 at 0:56
• Stillwell's is quite nice especially if one wants to understand quickly but still deeply enough the perennial newbie question "why/how do quaternions represent rotations?". Basically the first two chapters go deep enough for that. The first chapter has a calculation-based proof with some geometric intuition like but doesn't explain for instance why conjugation is used. The 2nd chapter puts enough group theory behind that to elucidate the matter. Lie algebras and the exponential map round up the picture in chapter 4. [continued] – Fizz Apr 12 '15 at 13:40
• The rest of the book transitions to more advanced material, IMHO mainly of interest to pure maths, like structure theory of Lie algebras. The book also contains a crash course on topology in chapter 8, which may be of independent interest. – Fizz Apr 12 '15 at 13:44

One of the main points of interest with regard to Lie group is their representations, and I think studying them together with their representations makes a lot of sense.

To this end, I recommend Fulton and Harris's book on representation theory. About 3/4 of it is devoted to Lie groups, and it light on the theoretical background (although it does presume some mathematical maturity) and heavy on examples and intuition.

• Dear Matthew, Fulton and Harris's book has some gaps that are not easy to 'fill in' for a first time learner. See for example (mathoverflow.net/questions/54561/…). Could you give some other references on Lie groups? – Bombyx mori Sep 12 '12 at 1:23
• @user32240: Dear user, To the extent that I know the subject at all, I learned Lie groups somewhat late in my mathematical career (after my post-doc), and I began with Fulton and Harris and then quickly moved onto more advanced texts (the Corvalis proceedings on automorphic forms, some of Knapp's books on unitary repreresentations). At earlier points in my studies I had tried to learn Lie theory from other texts (e.g. Jacobson's book on Lie algeras, and books by Helgason and Warner), without much success, whereas I found Fulton and Harris direct and enlightening. So I am a bit of an ... – Matt E Sep 12 '12 at 1:59
• ... F&H diehard, and don't really have any other more basic recommenations; but I'm sure some of the other books mentioned in the other answers have their merits too. Regards, – Matt E Sep 12 '12 at 2:00
• Dear Matthew, this is totally understandable and thank you a lot for mentioning other reference books. I only regret I could not read "between the lines" as Harvard graduate students, for which F&H is the targeted audience. Thanks. – Bombyx mori Sep 12 '12 at 2:22

Personally, seeing as you are a high school student, I would start out a little lighter--even lighter than Brian C. Hall's book.

I think the perfect place to get a painless introduction to Lie theory, that gives you the exact right idea without all of the necessary machinery is the little gem of a book "Matrix Groups for Undergraduates" by Tapp. There you will be introduced, in a very congenial and pleasant way, to Lie groups and the ideas of differential geometry simultaneously.

Once you get used to that I would suggest the book by Brian C. Hall that others have mentioned as well as the books by Sepanski and Tom Dieck. In fact, these are the recommended books for the Lie groups part of a course on Lie Groups/Algebraic Groups I'm taking with Jeffrey Adams (one of the big players in the discovery the article you linked to mentioned). You should get a good feel for compact Lie groups before you move onto the more advanced methods needed to discuss non-compact Lie groups.

Also, the notes by Ban and the accompanying lectures are great once you feel prepared to learn about non-compact Lie groups.

Also, an absolutely must read, for when you start learning the more advanced (i.e. anything beyond Tapp's book) topics in Lie groups is the fantastic introductory article Very Basic Lie Theory by Howe.

• Tapp is a good alternative to Stillwell's book and about on the same level of difficulty. However Tapp's book is slower-paced. One doesn't quite grasp how quaternions really work to represent rotations until the end of Tapp's book. Stillwell's book approach is more along the lines of "multiple passes, each revealing some more" instead of a build-up for the big finale. – Fizz Apr 12 '15 at 13:56

Lie groups are groups (obviously), but they are also smooth manifolds. Therefore, they usually come up in that context. If you want to learn about Lie groups, I recommend Daniel Bump's Lie groups and Anthony Knapp's Lie groups beyond an Introduction. But be aware that you need to know about smooth manifolds before delving into this topic; knowledge of basic group theory is not enough.

Also, as Adam Saltz noted boelow in the comments, if you want a book that treats both smooth manifolds and Lie groups, you can look at John Lee's Introduction to Smooth manifolds

• So i need to know about differential manifolds right? I can learn that by reading a book on differential geometry right? – Jorge Fernández Hidalgo Sep 12 '12 at 0:33
• @Khromonkey Well, it depends. Not all books on differential geometry will mention smooth manifolds in general. For this particular topic, I recommend John Lee's Introduction to Smooth manifolds: books.google.ca/books/about/… – M Turgeon Sep 12 '12 at 0:38
• In fact, Lee's book discusses lie groups! (try typing that ten times fast) – Adam Saltz Sep 12 '12 at 0:45
• @AdamSaltz Indeed! Thank you for pointing this out! – M Turgeon Sep 12 '12 at 0:46
• I want to point out that there's a second edition of my Introduction to Smooth Manifolds that just became available. The first edition is less expensive (if you buy it in paperback), but the second is a lot better. I don't mean to be advertising, but I just wanted to make sure everyone knows there's a newer edition before you decide what to buy. – Jack Lee Sep 13 '12 at 1:10

An introductory book in abstract algebra (at the same level of Dummit Foote) that does discuss the basic ideas of Lie Algebras (in a beautiful and not too technical way) is Michael Artin's Algebra.

Check it out!

I'd say Chevalley's book "Theory of Lie Groups I" is a good reference. I'm currently using him (yes, I'm studying Lie Groups too!). Take a look at it and see if it is what you need.

• This is a very good book, however readers should be warned that the book uses some terminology that isn't used anymore nowadays. This can be confusing to readers nowadays, see e.g. your question :) – t.b. Sep 12 '12 at 2:49

There is a modern book on Lie groups, namely

"Structure and Geometry of Lie Groups" by Hilgert and Neeb.

It is a lovely book. It starts with matrix groups, develops them in great details, then goes on to do Lie algebras and then delves into abstract Lie Theory. Although they develop the requisite differentiable manifold theory in the text, I would also suggest

"An Introduction to Manifolds" by Loring W. Tu

for the manifolds part.

I also endorse

"Lie Groups, Lie Algebras, and Representations" by Brian C. Hall

for an elementary introduction to matrix Lie groups.

• The first author is called Hilgert, not Hilbert. – Tobias Kildetoft Aug 5 '13 at 18:51

The last few sections of Teleman's representation theory notes are on the representation theory of the unitary group. I found them to be quite interesting, and a good introduction to Lie groups without Lie algebras. They won't get you to E_8, but they're still a good way to get into the subject if you already understand finite groups and their representations.

There have been a lot of terrific recommendations above(below?),but my favorite book on the subject hasn't been mentioned yet: Claudio Procesi's Lie Groups: An Approach through Invariants and Representations. Not only is it by one of the world's most respected researchers on the subject, it's probably the single most gentle book on the subject,even more so then Hall's book. The prerequisites are basically linear algebra and some rigorous calculus-everything else, including the concepts of differential manifolds, topology,tensor algebra and representation theory, are developed as needed in the book. It's very well written with a lot of strong exercises-to me,it's the best book for self study on the subject.

For students who don't have the patience to read through Procesi, there's a wonderful short chapter at the end of E. Vinberg's A Course In Algebra. It's gentle,builds on many concrete examples and gives the bare minimum students need to know.Also,as I've said many times before, I recommend Vinberg as probably my favorite single reference for algebra. Everyone serious about learning algebra should have a copy.

• Procesi's book can be intimidating for a beginner. – user38268 Sep 12 '12 at 22:28
• Although Procesi's book appeared in Springer's undergraduate-oriented Universitext series, I think it's alas not a serious contender for a first book on Lie groups for undergraduates next to those of Stillwell, Tapp, or Pollatsek. Any of these three manages to motivate the topic better and is substantially more accessible IMHO. – Fizz Apr 12 '15 at 14:12
• As another indicator of its actual difficulty, Procesi's book is the only (nominally) undergraduate book listed at www2.math.umd.edu/~jda/744 – Fizz Apr 12 '15 at 14:26

There is a nice book called Matrix Groups — An Introduction to Lie Group theory by Andrew Baker. It starts by talking on Matrix groups, then introduces Lie groups and shows that Matrix groups are in fact Lie groups. The last part is dedicated to the study of compact connected Lie groups. Note that it does not cover any representation theory. It is nice to read imho. A nice plus is that it repeats known results that are used. It really only assumes you have heard basic courses in linear algebra and analysis.

Another book that I actually liked a bit more is Naive Lie Theory from Stilwell. It is very easy to read and sometimes even a bit funny. It assumes only very, very basic mathematical knowledge and I recommend to read this to anyone who has never heard anything about that matter as a first read. I went through this book in two days, unable to put it down.

The book Lie Groups, Lie Algebras, and Representations – An Elementary Introduction from Brian Hall is a good book, as well. It doesn't read as good, but it seems to be nice as a reference book.

Another introductory book is Lie groups and algebras with applications to physics, geometry, and mechanics by Sattinger and Weaver.

• Welcome to Math.SE. If you have some personal experience with this book, some additional thoughts about its difficulty, breadth, etc. would be useful to future Readers. – hardmath Aug 25 '18 at 5:02
• IN my opinion this is a very accessible book with plenty of examples. – mo-user Aug 25 '18 at 5:12

An Introduction to the Lie Theory of One-Parameter Groups https://www.forgottenbooks.com/.../AnIntroductiontotheLieTheoryofOneParameterGr... Author: Abraham Cohen; Category: Calculus; Length: 256 Pages; Year: 1911.