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.
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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.
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- You can't put this one down!
- Two Thumbs Up
- GREAT READING FOR THE WHOLE FAMILY
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Ergodic Theory and Semisimple Groups (Monographs in Mathematics)
R.J. Zimmer
Manufacturer: Birkhäuser Boston
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Ergodic Theory (Cambridge Studies in Advanced Mathematics)
ASIN: 0817631844 |
Customer Reviews:
You can't put this one down!.......2004-01-07
The greatest book ever! Robert J. Zimmer captures the greatest math ever in this book! I read a book resently called To Kill A Mockingbird and it doesnt compare in any way to this perfect book. This book is a must read by all. You cant put this one down! Read it! Its great!
Two Thumbs Up.......2001-06-09
In all my years of studying mathematics, I have never encountered a more comprehensive text on semisimple groups and ergodic theory. I highly recomend this book to everyone from the amateur math lover to the greatest mathematician. Two thumbs up.
GREAT READING FOR THE WHOLE FAMILY.......2000-09-01
This book is one of the best I've ever read on semisimple groups. It even rivals G. A. Margulis's "Discrete Subgroups of Semisimple Lie Groups" as the best book I have ever read on semisimple groups. Furthermore, Zimmer's insight into Ergodic Theory is truely revolutionary. The entire book is great for all purposes, especially bedtime reading. Also, make sure to read it to to the kids; they'll love it. So, if you want the ver-i-tas about semisimple lie groups, ingeneous insights into ergodic theory, and great reading for the whole family - especially the kids, then make sure to read one of the great classics of the twentieth century: "Ergodic Theory and Semisimple Groups" by Robert J. Zimmer!
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- An excellent introduction to the topic of real simple Lie algebras
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Lectures on Real Semisimple Lie Algebras and Their Representations (ESI Lectures in Mathematics & Physics)
Arkady L. Onishchik
Manufacturer: Amer Mathematical Society
ProductGroup: Book
Binding: Paperback
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ASIN: 3037190027 |
Book Description
In 1914, E. Cartan posed the problem of finding all irreducible real linear Lie algebras. Iwahori gave an updated exposition of Cartan's work in 1959. This theory reduces the classification of irreducible real representations of a real Lie algebra to a description of the so-called self-conjugate irreducible complex representations of this algebra and to the calculation of an invariant of such a representation (with values $+1$ or $-1$) which is called the index. Moreover, these two problems were reduced to the case when the Lie algebra is simple and the highest weight of its irreducible complex representation is fundamental. A complete case-by-case classification for all simple real Lie algebras was given in the tables of Tits (1967). But actually a general solution of these problems is contained in a paper of Karpelevich (1955) that was written in Russian and not widely known.
The book begins with a simplified (and somewhat extended and corrected) exposition of the main results of Karpelevich's paper and relates them to the theory of Cartan-Iwahori. It concludes with some tables, where an involution of the Dynkin diagram that allows for finding self-conjugate representations is described and explicit formulas for the index are given. In a short addendum, written by J. V. Silhan, this involution is interpreted in terms of the Satake diagram.
The book is aimed at students in Lie groups, Lie algebras and their representations, as well as researchers in any field where these theories are used. Readers should know the classical theory of complex semisimple Lie algebras and their finite dimensional representation; the main facts are presented without proofs in Section 1. In the remaining sections the exposition is made with detailed proofs, including the correspondence between real forms and involutive automorphisms, the Cartan decompositions and the conjugacy of maximal compact subgroups of the automorphism group.
Published by the European Mathematical Society and distributed within the Americas by the American Mathematical Society.
Customer Reviews:
An excellent introduction to the topic of real simple Lie algebras.......2006-04-23
The main purpose of these notes is to give a self-contained and complete exposition of the representation theory of real semisimple Lie algebras. Although various texts on the topic exist, the originality of this small book is the elegance in the exposition and the presentation of some important facts that are absent in other treatises or only enumerated without further comment. Written by a prestigious expert in Lie theory, the text only demands a standard knowledge in the theory of complex Lie algebras and groups, and constitutes therefore an excellent text as a complement to an advanced course on the classification of complex semisimple Lie algebras and their representation theory.
The problem of classifying real simple and semisimple Lie algebras and their representations arises from the geometry of homogeneous spaces, and the first results in this direction were developed by E. Cartan himself in 1914. Using the more standardized algebraic theory and the work of Weyl, the study of real simple Lie algebras and groups was later expanded by various authors in order to develop a self-contained theory in analogy to the complex case. This work accomplishes this objective perfectly, and also pays homage to the important work of the late Fridrikh Izrailevich Karpelevich , who already solved many problems in the representation theory of real simple Lie algebras. However, these papers are unfortunately not widely known in the literature, and various of his results were later rediscovered by other authors.
The text is divided into nine sections, which present the main results with detailed proofs and illustrated with examples using the special simple algebra sl(n,C). The choice of this algebra is justified by the role it plays in the characterization of self-dual complex irreducible representations of real forms. For the remaining algebras the reader is led to the references.
The first section reviews the classical theory of semisimple complex Lie algebras, and fixes the notation that will be used in later chapters. The main material on compact groups that will be applied in the obtainment of real forms is also briefly presented, such as the theorem of Weyl. As recopilatory chapter, no proofs are given at this stage.
The second section deals with the complexification and realification of real and complex Lie algebras, respectively. Two important examples of real forms of complex semisimple Lie algebras are introduced: the real normal form, which can intuitively be interpreted as the algebra obtained by restriction of scalars and the compact form, which will be central for the construction of the remaining non-compact real forms. The first structural results concerning real forms are presented, namely, that real forms of simple complex algebras are simple, while complexification of real simple algebras are either simple or semisimple complex algebras [the insertion of the classical Lorentz algebra would have been welcomed after example 4]. The third section introduces the main tool used in the classification of real forms, the involutive automorphisms of a complex semisimple algebra and its correspondence with the real forms. It follows in particular that the compact form is unique. In order to describe this correspondence, the next section is devoted to various technical results concerning the automorphisms of complex semisimple algebras. Endowed with this machinery, the Cartan decomposition is discussed in detail. The conjugacy theorem of maximal compact subgroups of the adjoint linear group Int(g) is proved. Section 6 is devoted to an important problem which often appears in representation theory: given a homomorphism of complex semisimple Lie algebras f: ĝ →ĥ, which real forms of ĥ contain the image by f of some real form of ĝ? A satisfactory answer to this problem is given by means of the involutive automorphisms corresponding to the real forms. The material of this section follows the original work of F. I. Karpelevich in the beginning fifties . The material previously developed in chapter 3 concerning hermitean vector spaces is applied. Introducing a special class of morphisms, denoted S-homomorphisms , the result is sharpened. The seventh section devotes to the extension problem for irreducible representations for the case of the special linear Lie algebra sl(n,R). Special attention is devoted to the Karpelevich index, and the original formulae for computing this invariant are generalized to arbitrary involutive automorphisms. These results are applied in section 8 to classify explicitly the irreducible real representations in terms of highest weights, following the outline used by Iwahori in 1959. More precisely, real representations divide into two classes depending on the existence or not of an invariant complex structure. The last section, written by J. ?ilhan, presents an alternative classification by means of Satake diagrams, i.e., a generalization of the classical Dynkin diagram based on the introduction of two colors and arrows relating vertices of one color. It is described how to read off the involutions using these diagrams, and a characterization of self-dual complex irreducible representations is obtained. Additional material is presented in tabulated form at the end of this section.
Resuming, this book is very welcome reference on real simple Lie algebras, and has the innovation of presenting material that is distributed in many technical papers in a compact and effective way. It should be expected that this work will become a classic on the topic among the specialists in Lie algebras.
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Conjugacy Classes in Semisimple Algebraic Groups (Mathematical Surveys and Monographs)
James E. Humphreys
Manufacturer: American Mathematical Society
ProductGroup: Book
Binding: Hardcover
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ASIN: 0821803336 |
Book Description
After the fundamental work of Borel and Chevalley in the 1950s and 1960s on the structure and classification of semisimple algebraic groups over an arbitrary algebraically closed field, further results were obtained on conjugacy classes and centralizers. Conjugacy Classes in Semisimple Algebraic Groups draws together results achieved by Lusztig, Richardson, Spaltenstein, Springer, Steinberg, and others. This book provides a unified exposition of work done over the past thirty years.
Book Description
These notes, already well known in their original French edition, give the basic theory of semisimple Lie algebras over the complex numbers including the basic classification theorem. The author begins with a summary of the general properties of nilpotent, solvable, and semisimple Lie algebras. Subsequent chapters introduce Cartan subalgebras, root systems, and representation theory. The theory is illustrated by using the example of sln; in particular, the representation theory of sl2 is completely worked out. The last chapter discusses the connection between Lie algebras and Lie groups, and is intended to guide the reader towards further study.
Customer Reviews:
One of the most valuable expositions in Lie-theory.......2003-10-27
This book is intended as a short concise overview of the theory of complex semisimple Lie algebras. Inspite of its small volume, this text is far from being of easy lecture, since it assumes the knowledge of some basic facts concerning Lie algebras, as well as associative algebras. Indeed the first chapters are a résumé, without proofs, of some basic theorems of Lie algebras. This concerns solvable and nilpotent Lie algebras, as well as some generic results on semisimple algebras (results that do not involve Cartan subalgebras).
The proper exposition begins with the third chapter, dealing with Cartan subalgebras. Two fundamental facts are exposed in this chapter: existence and conjugacy of these subalgebras. The existence is proved by exhibiting the classical construction by means of regular elements, i.e., elements of the algebra whose annihilator is of minimal dimension. The conjugacy of Cartan subalgebras, which enables us to define the numeric invariant called rank, is developed in analogous way to the book of Chevalley [Théorie des Groupes de Lie, 1951]. Chapter four is devoted to the study of the complex simple Lie algebra of rank one, sl(2,C). This algebra plays the key role in the study of semisimple algebras and their representations, which justifies a separated treatment. The irreducible representations of sl(2,C) are obtained.
The root theory is introduced in the following chapter. Here the first innovation is made, namely, developing the root systems before dealing with the Cartan decomposition. In particular, no inner product has been used yet. Root systems are defined over a real vector space V, and the Weyl group is defined as the group generated by certain involutions associated to the roots [one will observe observe the similarity of this definition and the theory of Coxeter groups]. The inner product on V is obtained as an inner product which is invariant under the Weyl group. Bases of roots and their elementary properties are developed, and how to go from a basis to another by emans of the Weyl group [it is supposed that the root system is reduced, for nonreduced systems see for example the sixth chapter of Bourbaki: Algèbres de Lie, Hermann 1967]. Then it follows the notion of Cartan matrix (obtained from the inner product previously defined), and the associated Dynkin diagram. All admissible Dynkin diagrams are enumerated, and their corresponding root systems enumerated. Chapter six begin with the classical Weyl theorems, and the Cartan decomposition of a semisimple Lie algebra is obtained. From this the root system associated to the algebra follows naturally. The core of the chapter is the existence and uniqueness proofs of semisimple Lie algebras corresponding to a root system. As an appendix, a theorem showing how to construct semisimple Lie algebras from root systems by means of generators and relations [that is, using presentations]. This result is of extreme importance, and constitutes one of the germs that lead to the notion of Kac-Moody algebras in 1968. The next chapter is a standard treatment of representation theory of semisimple Lie algebras. The existence of dominant weights is shown, from which the (canonical) bijection between dominant integral forms and the finite dimensional irreducible modules follows. The Weyl character formula is also presented, but without proof.
The final chapter presents some important results of compact groups, intended to facilitate the lecture of more advanced texts [like the book of Pontryaguin, for example].
Resuming, an excellent text that presents a good insight to the theory of complex semiimple algebras. It should however be said that probably this book is not convenient for a first contact with these structures, due to its comprised presentation and the complete absence of exercises or many examples [this applies at least to the original french edition]. Actually, some acquaitance with Lie theory is implictly supposed through the text.
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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Dynamical Systems and Semisimple Groups: An Introduction (Cambridge Tracts in Mathematics)
Renato Feres
Manufacturer: Cambridge University Press
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ASIN: 0521591627 |
Book Description
Here is an introduction to dynamical systems and ergodic theory with an emphasis on smooth actions of noncompact Lie groups. The main goal is to serve as an entry into the current literature on the ergodic theory of measure preserving actions of semisimple Lie groups for students who have taken the standard first year graduate courses in mathematics. The author develops in a detailed and self-contained way the main results on Lie groups, Lie algebras, and semisimple groups, including basic facts normally covered in first courses on manifolds and Lie groups plus topics such as integration of infinitesimal actions of Lie groups. He then derives the basic structure theorems for the real semisimple Lie groups, such as the Cartan and Iwasawa decompositions and gives an extensive exposition of the general facts and concepts from topological dynamics and ergodic theory, including detailed proofs of the multiplicative ergodic theorem and Moore's ergodicity theorem. This book should appeal to anyone interested in Lie theory, differential geometry and dynamical systems.
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Equivariant Analytic Localization of Group Representations
Laura Smithies
Manufacturer: American Mathematical Society
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Binding: Mass Market Paperback
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ASIN: 0821827251 |
Book Description
The purpose of this book is to give a thorough treatment of the harmonic analysis of spherical functions on symmetric spaces. The theory was originally created by Harish-Chandra in the late 1950's and important additional contributions were made by many others in the succeeding years. The book attempts to give a definite treatment of these results from the spectral theoretic viewpoint. The harmonic analysis of spherical functions treated here contains the essentials of large parts of harmonic analysis of more general functions on semisimple Lie groups. Since the latter involves many additional technical complications, it will be very illuminating for any potential student of general harmonic analysis to see how the basic ideas emerge in the context of spherical functions. With this in mind, an attempt has been made only to use those methods (as far as possible) which generalize. Mathematicians and graduate students as well as mathematical physicists interested in semisimple Lie groups, homogeneous spaces, representations and harmonic analysis will find this book stimulating.
Book Description
Now in paperback, this graduate-level textbook is an excellent introduction to the representation theory of semi-simple Lie groups. Professor Varadarajan emphasizes the development of central themes in the context of special examples. He begins with an account of compact groups and discusses the Harish-Chandra modules of SL(2,R) and SL(2,C). Subsequent chapters introduce the Plancherel formula and Schwartz spaces, and show how these lead to the Harish-Chandra theory of Eisenstein integrals. The final sections consider the irreducible characters of semi-simple Lie groups, and include explicit calculations of SL(2,R). The book concludes with appendices sketching some basic topics and with a comprehensive guide to further reading. This superb volume is highly suitable for students in algebra and analysis, and for mathematicians requiring a readable account of the topic.
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Lie Groups: Beyond an Introduction
Lie Algebras of Finite and Affine Type (Cambridge Studies in Advanced Mathematics)
Unitary Representations of Reductive Lie Groups. (AM-118) (Annals of Mathematics Studies)
An Introduction to Homological Algebra (Cambridge Studies in Advanced Mathematics)
An Introduction to the Langlands Program
