File Name: gilmore lie groups lie algebras and some of their applications .zip
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MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up. 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. 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. 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. 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! 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.
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. The book "Introduction to Lie groups and Lie algebras" by A. Kirillov, Jr.
It might be a good starting point, and it has an excellent annotated bibliography. Edit: On further inspection, the. The actual book has the good bibliography. 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. 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.
If the material was covered in the Spring Lie groups course at Berkeley, then I prefer the presentation in this guy's notes.
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. 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:. 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. At any rate, it goes into more detail. 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. As an elementary introduction with lots of examples you may take a look at A. Baker,"Matrix Groups. After this a very good book with lot of results and almost self-contained, but rather demanding is M.
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.. Very fun, as always with Godement. 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. In my opinion, the best quick introduction to Lie group and algebra theory is in chapter 12 of E.
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.
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. A very down to earth introduction with many examples and clear explanations. Especially targeted at physicists, engineers and chemists. 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. His research areas include group theory, catastrophe theory, atomic and nuclear physics, singularity theory, and chaos. There are many courses, including something about Lie groups at J.
Milne's page: jmilne. If you read french: R. Another nice introductory book with many examples is Lie groups and algebras with applications to physics, geometry, and mechanics by Sattinger and Weaver. Sign up to join this community. The best answers are voted up and rise to the top.
Learning about Lie groups Ask Question. Asked 11 years, 5 months ago. Active 2 years, 6 months ago. Viewed 18k times. Improve this question. Charles Matthews Daniel Erman Daniel Erman 2, 2 2 gold badges 22 22 silver badges 23 23 bronze badges.
It does use a lot of analysis though a lot for me, anyway. Add a comment. Active Oldest Votes. Improve this answer. Also, the proofs are sometimes sketchy, so care should be exercised. Victor Protsak Victor Protsak Caleb Cheek Caleb Cheek 1 1 silver badge 4 4 bronze badges. Hall's really more for physics majors,but it's a nice book nevertheless.
J W J W 8 8 silver badges 19 19 bronze badges. Sam Lichtenstein Sam Lichtenstein 1, 1 1 gold badge 9 9 silver badges 16 16 bronze badges. It's incomplete, but gives one a good preview of the print version.
Anton Geraschenko Anton Geraschenko 22k 14 14 gold badges silver badges bronze badges. 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. Kevin H. Lin Kevin H. Lin I had not seen that before. Bischof Dec 18 '09 at
Add to Wishlist. By: Robert Gilmore. Product Description Product Details Lie group theory plays an increasingly important role in modern physical theories. Many of its calculations remain fundamentally unchanged from one field of physics to another, altering only in terms of symbols and the language. Using the theory of Lie groups as a unifying vehicle, concepts and results from several fields of physics can be expressed in an extremely economical way. With rigor and clarity, this text introduces upper-level undergraduate students to Lie group theory and its physical applications.
The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group—Lie algebra correspondence. The subject is part of differential geometry since Lie groups are differentiable manifolds. Lie groups evolve out of the identity 1 and the tangent vectors to one-parameter subgroups generate the Lie algebra. The structure of a Lie group is implicit in its algebra, and the structure of the Lie algebra is expressed by root systems and root data.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. I am looking for some material e. My motivation is that I eventually want to understand the theory underpinning papers such as these. The problem is, I am at the Rumsfeldian stage where I don't know what I don' t know.
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Par baca todd le mercredi, mars 30 , - Lien permanent. Download Lie groups, Lie algebras and some of their applications. In the same sense that the two books on the calculus of variations, Elsgolc , and W.
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