Lie Groups: Beyond an Introduction [Hardcover]

Anthony W. Knapp (Author)

Book Description

Publication Date: August 21, 2002 | ISBN-10: 0817642595 | ISBN-13: 978-0817642594 | Edition: 2nd

This book takes the reader from the end of introductory Lie group theory to the threshold of infinite-dimensional group representations. Merging algebra and analysis throughout, the author uses Lie-theoretic methods to develop a beautiful theory having wide applications in mathematics and physics. The book initially shares insights that make use of actual matrices; it later relies on such structural features as properties of root systems.

Editorial Reviews

Review

"Anthony Knapp's Lie Groups Beyond an Introduction, 2nd edition, is a beautiful introduction to this area of mathematics, appropriate for a variety different audiences.... The book is well-organized with concise, focused introductions to each chapter, a very thorough index of notation, and appendices.... In addition, there are hints to the hundreds of exercises, and a section on historical notes.... Knapp's writing is clear, and he avoids excessive notation. The first few chapters comprise a standard introductory course in Lie theory, while numerous second courses could be taught out of the later chapters. Its breadth of coverage and extensive tables also make the book a valuable reference for researchers in representation theory."   —MAA Reviews (review of the second edition) "The first edition of the present book appeared in 1996, and quickly became one of the standard references on the subject…. The present edition has been perfected even further, apart from straightening occasional errors…and making various revisions throughout, by adding a new introduction and two new chapters [IX and X]…. Chapter IX contains a treatment of induced representations and branching theorems…. Chapter X is largely about actions of compact Lie groups on polynomial algebras, pointing toward invariant theory and some routes to infinite-dimensional representation theory…. This is an excellent monograph, which, as with the previous edition, can be recommended both as a textbook or for reference to anyone interested in Lie theory."   —Mathematical Bohemica (review of the second edition) "The important feature of the present book is that it starts from the beginning (with only a very modest knowledge assumed) and covers all important topics.... The book is very carefully organized [and] ends with 20 pages of useful historic comments. Such a comprehensive and carefully written treatment of fundamentals of the theory will certainly be a basic reference and text book in the future."   —Newsletter of the EMS (review of the first edition) "Each chapter begins with an excellent summary of the content and ends with an exercise section.... This is really an outstanding book, well written and beautifully produced. It is both a graduate text and a monograph, so it can be recommended to graduate students as well as to specialists."   —Publicationes Mathematicae (review of the first edition) "This is a wonderful choice of material. Any graduate student interested in Lie groups, differential geometry, or representation theory will find useful ideas on almost every page. Each chapter is followed by a long collection of problems [that] are interesting and enlightening [and] there are extensive hints at the back of the book. The exposition...is very careful and complete.... Altogether this book is delightful and should serve many different audiences well. It would make a fine text for a second graduate course in Lie theory."   —Bulletin of the AMS (review of the first edition) "This is a fundamental book and none, beginner or expert, could afford to ignore it. Some results are really difficult to be found in other monographs, while others are for the first time included in a book."   —Mathematica (review of the first edition) "The book is written in a very clear style, with detailed treatment of many relevant examples. . . . The book is eminently suitable as a text from which to learn Lie theory."   —Mathematical Reviews (review of the first edition) "The important feature of the present book is that it starts from the beginning (with only a very modest knowledge assumed) and covers all important topics... The book is very carefully organized [and] ends with 20 pages of useful historic comments. Such a comprehensive and carefully written treatment of fundamentals of the theory will certainly be a basic reference and textbook in the future." —Newsletter of the EMS (review of the first edition) "It is a pleasure to read this book. It should serve well different audiences. It perfectly suits as a text book to learn Lie theory, including Lie groups, representation theory, and structure theory of Lie algebras. The absense of misprints and errors as well as the long collection of problems including hints at the back of the book make it suitable for self-study. . . At the end there are a lot of enlightening historical remarks, references, and additional results which can serve as a guide for further reading.  The book has two good indices. Specialists will be able to use it as a reference for formulation and proofs of the basic results but also for details concerning examples of semisimple groups." ---ZAA

From the Back Cover

Lie Groups Beyond an Introduction takes the reader from the end of introductory Lie group theory to the threshold of infinite-dimensional group representations. Merging algebra and analysis throughout, the author uses Lie-theoretic methods to develop a beautiful theory having wide applications in mathematics and physics. A feature of the presentation is that it encourages the reader's comprehension of Lie group theory to evolve from beginner to expert: initial insights make use of actual matrices, while later insights come from such structural features as properties of root systems, or relationships among subgroups, or patterns among different subgroups. Topics include a description of all simply connected Lie groups in terms of semisimple Lie groups and semidirect products, the Cartan theory of complex semisimple Lie algebras, the Cartan-Weyl theory of the structure and representations of compact Lie groups and representations of complex semisimple Lie algebras, the classification of real semisimple Lie algebras, the structure theory of noncompact reductive Lie groups as it is now used in research, and integration on reductive groups. Many problems, tables, and bibliographical notes complete this comprehensive work, making the text suitable either for self-study or for courses in the second year of graduate study and beyond.

Product Details

• Hardcover: 830 pages
• Publisher: Birkhäuser Boston; 2nd edition (August 21, 2002)
• Language: English
• ISBN-10: 0817642595
• ISBN-13: 978-0817642594
• Product Dimensions: 9.6 x 6.2 x 1.7 inches
• Shipping Weight: 3 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #248,888 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

71 of 73 people found the following review helpful:

5.0 out of 5 stars Review of Knapp's "Lie groups: beyond an introduction.", August 12, 2002

By 

Howard Barnum (New Mexico) - See all my reviews

This review is from: Lie Groups Beyond an Introduction (Progress in Mathematics) (Hardcover)

The short version: this is a superbly written and conceived book; if I had to learn this material (the basic theory of
structure and representation of Lie algebras and groups,
especially semimsimple ones) from a single book, this is
the one I'd choose, among those I've seen. If you know the
basics of abstract algebra and some very basic concepts from
topology and manifolds, and you want to learn this material,
use this book. It would be a good reference, too, as it is
easy to find things in it, and takes a fairly modern, sophisticated approach (without sacrificing motivation and
intuition).

The long version, if you want more convincing or details:

I have used several books recently in learning the structure and
representation theory of Lie algebras and groups (especially Humphreys' Introduction to Lie algebras and representation theory, Fulton
and Harris' "Representation Theory," Varadarajan's "Lie groups,
Lie algebras, and their representations.") Although I came to Knapp's book with a decent background from the others, I think it's the best pedagogically, for someone with a modicum of mathematical sophistication and some basics like abstract
algebra and an idea of what a smooth manifold is), and a smattering of Lie theory. Some examples of the book's strength:
Elementary but potentially confusing concepts (like complexification, real forms, field extensions)
are explained thoroughly but in a sophisticated way, rather
than viewed as obvious. Carefully chosen examples motivate and
clarify the general theory; consequently even though the book
is completely rigorous, and carefully delineates lemmas, proofs,
remarks, definitions, and the like, it seems less dry then some
others (e.g. Varadarajan, from my point of view). But the point
of the examples, and their relation to the general theory, is
made clear, so they do not provide an overload of detail or b
obscure the main structure. Thought is always given to the
reader's understanding, not just to logical correctness, though
the author also takes the point of view, with which I concur,
that logical clarity and sufficient detail are essential
to understanding. Relations between ideas, alternative
proofs, and the structure of the theory to come are discussed
thoroughly, but such discussion is clearly demarcated from
the main structure of the argument, so that the latter is never
obscured. This is a fantastic book, and exactly what I was
looking for. Whether you are learning the material for the
first time, or want to review it or refer to, it is a superb
source.