Book Description
From reviews 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 text book in the future." -- Newsletter of the EMS "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 "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 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.
Customer Reviews:
Review of Knapp's "Lie groups: beyond an introduction.".......2002-08-13
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.
Book Description
In this classic work, Anthony W. Knapp offers a survey of representation theory of semisimple Lie groups in a way that reflects the spirit of the subject and corresponds to the natural learning process. This book is a model of exposition and an invaluable resource for both graduate students and researchers. Although theorems are always stated precisely, many illustrative examples or classes of examples are given. To support this unique approach, the author includes for the reader a useful 300-item bibliography and an extensive section of notes.
Customer Reviews:
Very detailed with lots of motivating examples.......2003-04-20
The theory of representations of semisimple Lie groups is very complete from a mathematical perspective and is of enormous importance in high energy physics. This book gives a comprehensive overview of this theory, and deals with both the noncompact and compact cases. My interest was with the noncompact case and in topics such as the Langland's classification, and so I read only chapters 5 - 10. Therefore my review will be confined to these chapters. Throughout the book, G denotes the group in question and K denotes the elements of G fixed under the Cartan involution. The author endeavors, and this is reflected in the title of the book, to employ many examples to illustrate the main results. This makes the book considerably more easy to follow than others that are written in the "Bourbaki" style.
The Iwasawa and Bruhat decompositions and the Weyl group construction are shown to hold for non-compact groups in chapter 5. The Borel-Weil theorem is proven for compact connected Lie groups using the results of the chapter. The Harish-Chandra decomposition fo linear connected reductive groups is proven in chapter 6. The author shows clearly the role of holomorphic representations in obtaining this result and the construction of holomorphic discrete series. The principal series representations of SL(2, R) and SL(2, C) are use to motivate the notion of an 'induced representation" in chapter 7. The theory of induced representations involves the Bruhat theory and its use of distribution theory, and relates via the 'intertwining operators', irreducible representations of two subgroups.
The author discusses the notion of an admissible representation in chapter 8, which are representations on a Hilbert space by unitary operators and each element in K has finite multiplicity when the representation is restricted to K. Equivalence of admissible representations are discussed via the concept of an "infinitesimal equivalance", which is the usual notion if the representation is unitary and irreducible. The Langlands classification of irreducible admissible representations is discussed in detail. The Langlands program shows to what extent irreducible admissible representations of a group are determined by the parabolic subgroups. The construction of discrete series, used throughout the proof of the Langlands classification, is then done in detail in the next chapter. Ths concept of an admissible infinitesimally unitary representation plays particular importance here. Here the representation operators act like skew-Hermitian operators with respect to an inner product on the space of K-finite vectors. If one reads this chapter from a physics perspective, the representations constructed using discrete series are somewhat 'exotic' and will probably not enter into applications, in spite of the fact that physical considerations do dictate sometimes the use of noncompact groups.
Chapter 10 addresses the question as to the completeness of irreducible admissible representations using discrete series. If there not enough discrete series representations this will show up in the Fourier analysis of square integrable functions on the group. In the compact case, Fourier analysis proceeded via the characters of irreducible representations. The author shows how to do this in the noncompact case via 'global characters' of representations, which are well-behaved generalizations of the compact case. The well-behavedness of global characters comes from their being of trace class, with the result of the trace being a distribution. The author gives explicit formulas for the case of SL(2, R), and shows hows differential equations can be used to limit the possibilities for how characters behave. In fact, the author shows to what extent characters are functions, proving that the restriction of any irreducible global character of G to the 'regular set' is a real analytic function.
Book Description
An introductory text book for graduates and advanced undergraduates on group representation theory. It emphasizes group theory's role as the mathematical framework for describing symmetry properties of classical and quantum mechanical systems.
Familiarity with basic group concepts and techniques is invaluable in the education of a modern-day physicist. This book emphasizes general features and methods which demonstrate the power of the group-theoretical approach in exposing the systematics of physical systems with associated symmetry.
Particular attention is given to pedagogy. In developing the theory, clarity in presenting the main ideas and consequences is given the same priority as comprehensiveness and strict rigor. To preserve the integrity of the mathematics, enough technical information is included in the appendices to make the book almost self-contained.
A set of problems and solutions has been published in a separate booklet.
Customer Reviews:
He can do better........2002-01-15
This is not a bad book. The contents are comprehensive. The theorems are well proved. Graphs are used appropriatly to clarify the concept. But it has several shortcomings. One is the line space is too small, too many words are clustered together. Another is sometimes the proofs are too short. Some explanation should be given on certain points because those points are not so obvious and should not be taken for granted. Frankly, I love group theory, but after reading this book for 20 pages, I felt very tired, both because of the bad format and hard thinking. Hamermesh's book will give you a more enjoyable reading.
Almost perfect!.......2000-06-16
This book gives an excellent introduction into group theory and provides a working knowledge for those who want to study field theory (for example). I especially like the complete treatment of representations of the classical groups and the Lorentz and Poincare groups.
Book Description
In this renowned volume, Hermann Weyl discusses the symmetric, full linear, orthogonal, and symplectic groups and determines their different invariants and representations. Using basic concepts from algebra, he examines the various properties of the groups. Analysis and topology are used wherever appropriate. The book also covers topics such as matrix algebras, semigroups, commutators, and spinors, which are of great importance in understanding the group-theoretic structure of quantum mechanics.
Hermann Weyl was among the greatest mathematicians of the twentieth century. He made fundamental contributions to most branches of mathematics, but he is best remembered as one of the major developers of group theory, a powerful formal method for analyzing abstract and physical systems in which symmetry is present. In The Classical Groups, his most important book, Weyl provided a detailed introduction to the development of group theory, and he did it in a way that motivated and entertained his readers. Departing from most theoretical mathematics books of the time, he introduced historical events and people as well as theorems and proofs. One learned not only about the theory of invariants but also when and where they were originated, and by whom. He once said of his writing, "My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful."
Weyl believed in the overall unity of mathematics and that it should be integrated into other fields. He had serious interest in modern physics, especially quantum mechanics, a field to which The Classical Groups has proved important, as it has to quantum chemistry and other fields. Among the five books Weyl published with Princeton, Algebraic Theory of Numbers inaugurated the Annals of Mathematics Studies book series, a crucial and enduring foundation of Princeton's mathematics list and the most distinguished book series in mathematics.
Customer Reviews:
Hard-core Group Theory.......2001-03-15
Forget about that pansy abstract axiom approach. This is the WWF of group theory. Weyl will take anyone to the mat with this book. It is packed with detail and demonstrations. He follows the vector space/matrix representation approach common to digital systems, physics and chemistry rather than axiomatic, generators/permutations approach more common in Abstract Algebra courses. This is the lineage that develops matrix transforms as groups starting from "the full group of all non-singular linear transformations and .. the orthogonal groups" (p. vii). The latter chapters cover characters and invariants. Galois and field theory have been vanquished. Chapter 2, "Remembrance of things past" is very entertaining. My favorite quote, "Here there is only one man to mention - Hilbert. His papers (1890/92) mark a turning point in the history of invariants theory. He solves the main problems and thus almost kills the whole subject." It's funny because it's true. This is almost a botanical treatise in which the matrix groups are studied as specimens in the jungle -- "...after all each group stands in its own right and does not deserve to be looked upon merely as a subgroup of...Her All-embracing Majesty GL(n)." (p 136). Historic references throughout provide motivation and entertainment. You couldn't possibly be disappointed with this book.
Great.......2000-04-27
Although this is a dated work, lacking some of the more modern language, it is still worth owning and reading. It is, after all, a designated "classic." And the material presented has been incorporated within so many aspects of physics that one simply cannot avoid needing a book such as this. There are better books on the subject, for both mathematicians and physicists, but this book still proves its worth.
Average customer rating:
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Kac-Moody Groups, Their Flag Varieties & Representation Theory
Shrawan Kumar , and
S. Kumar
Manufacturer: Birkhäuser Boston
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ASIN: 0817642277 |
Book Description
This is the first monograph to exclusively treat Kac-Moody (K-M) groups, a standard tool in mathematics and mathematical physics. K-M Lie algebras were introduced in the mid-sixties independently by V. Kac and R. Moody, generalizing finite-dimensional semisimple Lie algebras. K-M theory has since undergone tremendous developments in various directions and has profound connections with a number of diverse areas, including number theory, combinatorics, topology, singularities, quantum groups, completely integrable systems, and mathematical physics. This comprehensive, well-written text moves from K-M Lie algebras to the broader K-M Lie group setting, and focuses on the study of K-M groups and their flag varieties. In developing K-M theory from scratch, the author systematically leads readers to the forefront of the subject, treating the algebro-geometric, topological, and representation-theoretic aspects of the theory. Most of the material presented here is not available anywhere in the book literature. {\it Kac--Moody Groups, their Flag Varieties and Representation Theory} is suitable for an advanced graduate course in representation theory, and contains a number of examples, exercises, challenging open problems, comprehensive bibliography, and index. Research mathematicians at the crossroads of representation theory, geometry, and topology will learn a great deal from this text; although the book is devoted to the general K-M case, those primarily interested in the finite-dimensional case will also benefit. No prior knowledge of K-M Lie algebras or of (finite-dimensional) algebraic groups is required, but some basic knowledge would certainly be helpful. For the reader's convenience some of the basic results needed from other areas, including ind-varieties, pro-algebraic groups and pro-Lie algebras, Tits systems, local cohomology, equivariant cohomology, and homological algebra are included.
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The Fourfold Way in Real Analysis: An Alternative to the Metaplectic Representation (Progress in Mathematics)
Andre Unterberger
Manufacturer: Birkhauser
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ASIN: 3764375442 |
Book Description
The fourfold way starts with the consideration of entire functions of one variable satisfying specific estimates at infinity, both on the real line and the pure imaginary line. A major part of classical analysis, mainly that which deals with Fourier analysis and related concepts, can then be given a parameter-dependent analogue. The parameter is some real number modulo 2, the classical case being obtained when it is an integer. The space L2(R) has to give way to a pseudo-Hilbert space, on which a new translation-invariant integral still exists. All this extends to the n-dimensional case, and in the alternative to the metaplectic representation so obtained, it is the space of Lagrangian subspaces of R2n that plays the usual role of the complex Siegel domain. In fourfold analysis, the spectrum of the harmonic oscillator can be an arbitrary class modulo the integers. Even though the whole development touches upon notions of representation theory, pseudodifferential operator theory, and algebraic geometry, it remains completely elementary in all these aspects. The book should be of interest to researchers working in analysis in general, in harmonic analysis, or in mathematical physics.
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Relativity, Groups, Particles: Special Relativity and Relativistic Symmetry in Field and Particle Physics
Roman U. Sexl , and
Helmuth K. Urbantke
Manufacturer: Springer
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ASIN: 3211834435 |
Book Description
This textbook attempts to bridge the gap that exists between the two levels on which relativistic symmetry is usually presented – the level of introductory courses on mechanics and electrodynamics and the level of application in high-energy physics and quantum field theory: in both cases, too many other topics are more important and hardly leave time for a deepening of the idea of relativistic symmetry. So after explaining the postulates that lead to the Lorentz transformation and after going through the main points special relativity has to make in classical mechanics and electrodynamics, the authors gradually lead the reader up to a more abstract point of view on relativistic symmetry – always illustrating it by physical examples – until finally motivating and developing Wigner’s classification of the unitary irreducible representations of the inhomogeneous Lorentz group. Numerous historical and mathematical asides contribute to conceptual clarification.
Book Description
This book offers a systematic treatment--the first in book form--of the development and use of cohomological induction to construct unitary representations. George Mackey introduced induction in 1950 as a real analysis construction for passing from a unitary representation of a closed subgroup of a locally compact group to a unitary representation of the whole group. Later a parallel construction using complex analysis and its associated co-homology theories grew up as a result of work by Borel, Weil, Harish-Chandra, Bott, Langlands, Kostant, and Schmid. Cohomological induction, introduced by Zuckerman, is an algebraic analog that is technically more manageable than the complex-analysis construction and leads to a large repertory of irreducible unitary representations of reductive Lie groups.
The book, which is accessible to students beyond the first year of graduate school, will interest mathematicians and physicists who want to learn about and take advantage of the algebraic side of the representation theory of Lie groups. Cohomological Induction and Unitary Representations develops the necessary background in representation theory and includes an introductory chapter of motivation, a thorough treatment of the "translation principle," and four appendices on algebra and analysis.
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Lie Groups, Lie Algebras, and Representations: An Elementary Introduction
Representation Theory: A First Course (Graduate Texts in Mathematics / Readings in Mathematics)
Introduction to Lie Algebras and Representation Theory (Graduate Texts in Mathematics)
Representation Theory of Semisimple Groups: An Overview Based on Examples. (PMS-36).
Differential Geometry, Lie Groups, and Symmetric Spaces (Graduate Studies in Mathematics)
Theory of Complex Homogeneous Bounded Domains (Mathematics and Its Applications)
Noetherian Semigroup Algebras (Algebra and Applications)
Elements of Mathematics: Algebra I Chapters 1-3
Yetter-Drinfel'd Hopf Algebras over Groups of Prime Order (Lecture Notes in Mathematics)
Ideals and Reality: Projective Modules and Number of Generators of Ideals (Springer Monographs in Mathematics)
