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Rating: 
Summary: Very nice
Review: 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.
Rating:
Summary: Very nice
Review: 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.
Rating: 
Summary: Brilliantly Clear
Review: An excellent companion for anybody learning lie algebras or representation theory. Also good for physics folk needing to pick up more than the basics of lie algebras; a nice followup to a "lie algebras in physics" book (and there are many of those.)In particular, some people really need to buy this book.
Rating:
Summary: A beautiful exposition
Review: 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.
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