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The primary goal of these lectures is to introduce a beginner to the finite-dimensional representations of Lie groups and Lie algebras. Intended to serve non-specialists, the concentration of the text is on examples. The general theory is developed sparingly, and then mainly as useful and unifying language to describe phenomena already encountered in concrete cases. The book begins with a brief tour through representation theory of finite groups, with emphasis determined by what is useful for Lie groups. The focus then turns to Lie groups and Lie algebras and finally to the heart of the course: working out the finite dimensional representations of the classical groups. The goal of the last portion of the book is to make a bridge between the example-oriented approach of the earlier parts and the general theory.
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Given that over three-quarters of this book is devoted to the representation theory of Lie groups and Lie algebras, why have a discussion of the representations of finite groups at all? Read the first page Browse Sample Pages
Front Cover | Copyright | Table of Contents | Excerpt | Index | Back Cover
Given that over three-quarters of this book is devoted to the representation theory of Lie groups and Lie algebras, why have a discussion of the representations of finite groups at all? Read the first page Browse Sample Pages
Front Cover | Copyright | Table of Contents | Excerpt | Index | Back Cover
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7 reviews
41 of 41 people found the following review helpful
Very nice
12 Aug 2002
Format:Paperback
An excellent book. The approach, working toward the general theory via examples, has some great pedagogical virtues but also drawbacks. It also means the book has drawbacks as a reference, as important general theorems can be hard to locate (often they are in an appendix, but relevant definitions or lemmas are in several places in the text). Despite the example-oriented style, the level of mathematical sophistication assumed is reasonably high (so some physicists, for example, may find some of the explanations require boning up on certain ideas found more in pure mathematics than physics). However, many things are given very nice explanations that are lacking in some dryer texts (e.g. Varadarajan, or even Humphreys). Particularly nice is the discussion of relations between the representation theory of finite groups and Lie groups. Many mathematicians might find this book an enjoyable read to see connections made and examples worked out at a high level of sophistication, after learning the general theory. Some may also find it useful primarily as a repository of worked-out examples. I found Humphreys book "Introduction to Lie algebras and representation theory (Springer GTM series) to be an essential companion for getting the general theory with full proofs in a somewhat more logical order, if somewhat terse and a tad dry; Knapp's book "Lie groups beyond an introduction" could also serve this purpose, perhaps even somewhat better. If teaching a course, I would probably use this as supplemental reading rather than a primary text (though it could also turn out that gradually-generalizing-from-examples approach works better in a course than for self-teaching). It has been a useful book for me to own, and I recommend it, with the caution that you will probably want to supplement it with a book like Knapp's. (If you want to use only one book, and are reasonably mathematically sophisticated and already know basically what Lie groups and algebras are, use Knapp's.) I am a math-oriented physicist, who recently learned much of this material, using this and other books, in order to use it in my research.
28 of 29 people found the following review helpful
A beautiful exposition
6 Sep 2000
Format:Paperback
This is an absolutely delightful introduction to the theory of Lie groups and their representations. The style is informal but informative, with some of the important proofs hidden in the appendex or even omitted (i.e. existance of the finite dimensional representations for all lie algebras). However, this is a fully rigorous text, and all the important theorems are stated, and most are proved. Mathematicians should suppliment this book with Humphries standard text on Lie algebras. However, this book provides motivation and intuitive insight that Humphries is missing. Additional enjoyment may be derived from the sampling of other unusual topics, such as Schur functors and applications to algebraic geometry. Of course, these can also be omitted as the reader desires. Read a lecture every few nights before bedtime, and soon Lie theory will seem beautiful and almost intuitive.
9 of 10 people found the following review helpful
Thorough and strong in content; remarkably unfriendly
5 Mar 2011
Format:Paperback|Amazon Verified Purchase
This is the book that seems to have everything. Written by two of the masters with such ambition as to cover representations of finite groups, representations of Lie algebras, together with countless detailed examples (and many pictures to boot!), what could go wrong?
Within the first few pages, though, you should begin to feel that something is amiss. Proofs and arguments are almost always incomplete. Details are never provided under any circumstances. Example computations are beautiful and swift, but usually rely on an understanding that is deeper than actually presented in the text, using lemmas not present anywhere in the entire volume. They are the sorts of computations which, if included on a homework assignment and graded by someone well-versed in the subject, would get at most half the marks with several copies of the comment "yes, but you need to explain why." Nearly half the subject, you will realize after close analysis, is just left to the reader. The authors also supplement the instruction with an annoying delusion that the entire book is trivial; they will repeatedly tell you that everything here is trivial, easy, or immediate, but they will never acknowledge anything as being hard. Not only is this of course wrong, but it's disrespectful to their brilliant predecessors who toiled day and night to bring to them these apparently trivial truths.
This is an exceptionally dangerous book to learn from. It's the sort of book that makes you think you understand the details when in fact you have no idea what you're talking about. It makes you think something's trivial or simple when it actually requires some clever thinking. Given the book's length, it is clear that the authors were simply too ambitious. One (or, evidently, two!) cannot cover this range of material in appropriate detail and with due care to the reader without violating all reasonable restrictions on how fat and bloated any single volume should permit itself to become before giving into gluttonous sin.
This book isn't all bad, though. It makes a decent reference due to its ambition. There are some nice pictures. And the methods of computation really are nice--just don't think you understand them if you haven't written pages of extra notes filling in the gaps.
Vinberg's Linear Representations of Groups is a much superior treatment of the basics of the subject. After using Fulton and Harris's book, you may be surprised to see how much more space it takes Vinberb to cover what Fulton and Harris annihilate in a few pages or even paragraphs here. And then you will realize how frail and weak the treatment of individual topics actually is in the present book.
Within the first few pages, though, you should begin to feel that something is amiss. Proofs and arguments are almost always incomplete. Details are never provided under any circumstances. Example computations are beautiful and swift, but usually rely on an understanding that is deeper than actually presented in the text, using lemmas not present anywhere in the entire volume. They are the sorts of computations which, if included on a homework assignment and graded by someone well-versed in the subject, would get at most half the marks with several copies of the comment "yes, but you need to explain why." Nearly half the subject, you will realize after close analysis, is just left to the reader. The authors also supplement the instruction with an annoying delusion that the entire book is trivial; they will repeatedly tell you that everything here is trivial, easy, or immediate, but they will never acknowledge anything as being hard. Not only is this of course wrong, but it's disrespectful to their brilliant predecessors who toiled day and night to bring to them these apparently trivial truths.
This is an exceptionally dangerous book to learn from. It's the sort of book that makes you think you understand the details when in fact you have no idea what you're talking about. It makes you think something's trivial or simple when it actually requires some clever thinking. Given the book's length, it is clear that the authors were simply too ambitious. One (or, evidently, two!) cannot cover this range of material in appropriate detail and with due care to the reader without violating all reasonable restrictions on how fat and bloated any single volume should permit itself to become before giving into gluttonous sin.
This book isn't all bad, though. It makes a decent reference due to its ambition. There are some nice pictures. And the methods of computation really are nice--just don't think you understand them if you haven't written pages of extra notes filling in the gaps.
Vinberg's Linear Representations of Groups is a much superior treatment of the basics of the subject. After using Fulton and Harris's book, you may be surprised to see how much more space it takes Vinberb to cover what Fulton and Harris annihilate in a few pages or even paragraphs here. And then you will realize how frail and weak the treatment of individual topics actually is in the present book.
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