Book Description
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.
Customer Reviews:
Very nice.......2002-08-13
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.
A beautiful exposition.......2000-09-06
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.
Brilliantly Clear.......1999-11-22
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.
Book Description
This text is an introduction to the representation theory of the symmetric group from three different points of view: via general representation theory, via combinatorial algorithms, and via symmetric functions. It is the only book to deal with all three aspects of this subject at once. The style of presentation is relaxed yet rigorous and the prerequisites have been kept to a minimum¿undergraduate courses in linear algebra and group theory will suffice.
Customer Reviews:
Worth the price just for the first chapter.......2007-02-08
Sagans book makes representation theory easy. The book first covers representations using modules and then choosing a basis to show the matrix approach. With every new topic he develops it using what Doron Zeilberger has dubbed the Gelfand Principle ([...]) The principle is: "Always chooses the smallest example to make a point". It isn't easy to find the smallest example when Sn grows as quickly as it does, but Sagen always manages to do it.
The ensuing chapters follow in the same vein. Ideas are introduced and explained, sometimes with pictures, sometimes with calculations, but always as clearly as can be.
To read this book does require a firm grounding in linear algebra, as well as abstract algebra. Time reading it is time well spent.
Near Perfect.......2003-04-06
This book is excellent. The material is presented clearly and concisely. It makes the subject matter accessible and interesting. I used it as the text for a one-semester graduate subject. I completed all of the exercises, so it is well-paced for this kind of study. I started with only an introductory knowledge of group theory, so it is self-contained. The only drawback is that there are no solutions to any of the exercises. If it had this, it would be a perfect bok.
Good introduction for representation theory........2000-03-25
This book has 4 chapters.Chapter1 is about general theory of representations of finite group.Chapter2 is about representation of symmetric groups.chapter3 and 4 are about combinatorial topics and symmetric functions. Though I haven't read all of the book,I highly recommand this book because this book shows us introductive part of representation theory with easy words.I think it is worth to read for all who are to begin the study of representation theory.
Good introduction for representation theory........2000-03-25
This book has 4 chapters.Chapter1 is about general theory of representations of finite group.Chapter2 is about representation of symmetric groups.chapter3 and 4 are about combinatorial topics and symmetric functions. Though I haven't read all of the book,I highly recommand this book because this book shows us introductive part of representation theory with easy words.I think it is worth to read for all who are to begin the study of representation theory.
Book Description
This book's organizing principle is the interplay between groups and rings, where “rings” includes the ideas of modules. It contains basic definitions, complete and clear theorems (the first with brief sketches of proofs), and gives attention to the topics of algebraic geometry, computers, homology, and representations. More than merely a succession of definition-theorem-proofs, this text put results and ideas in context so that students can appreciate why a certain topic is being studied, and where definitions originate. Chapter topics include groups; commutative rings; modules; principal ideal domains; algebras; cohomology and representations; and homological algebra. For individuals interested in a self-study guide to learning advanced algebra and its related topics.
Customer Reviews:
excellent.......2006-03-30
This is a very clear introduction to graduate-level algebra. It is much better organized than Dummit and Foote. I particularly like the treatment of modules.
The worst mathematics book I have ever read!!!.......2004-07-21
I gave this book one star only because I couldn't give it a score of zero!!! Although many professors say that this book is excellent, remember they are professors who already understand the material. This book shows no examples, and the examples that it does show end abruptly with comments such as "all items are routine." Routine!!! Please show me what to do so that I don't have to spend more money on a separate study guide. Aren't mathematics texts expensive enough? This book may be an excellent addition to a professors library but this book should never, ever be used as a primary text for students.
Good for Self-Study.......2003-08-22
This is a tough book to review, because it is not clear who the real audience is supposed to be. The author says that it is aimed at first-year graduate students, with a bunch of extra material that can be referred back to during the second year and beyond. The earlier chapters also include efficient reviews (with sketched proofs) of material that should be familiar to those who have taken undergraduate algebra.
This characterization is debatable. Based on my experience reading most of the first six chapters (the first 400 out of about 1000 pages), I would say that the level of sophistication is roughly that of Dummit and Foote's "Abstract Algebra", which is usually considered an undergraduate book. D&F can sometimes be harder to read, and that is in part because Rotman's exposition is better (in my opinion), but also because D&F introduce more difficult material earlier. Whether D&F's approach is better is questionable; I find Rotman to be a much smoother read, but the organization is quite different -- for example, one does not encounter noncommutative rings until deep into the book, whereas Dummit and Foote introduce them immediately upon defining rings. On the other hand, early in the coverage of D&F's chapter on rings, one has to digest Zorn's Lemma and its applications almost from the beginning, whereas Rotman (I think wisely) pushes this back into a later section. In general, D&F introduce a lot of hairy examples that by themselves require a lot of effort to digest (thereby impeding the reader's progress through the core material), whereas Rotman's examples tend to be straightforward, at least as new concepts are being presented.
So, overall, the exposition flows more smoothly in Rotman's book, and the reader can cover the basics more quickly with less time spent on tangential examples and early generalizations. Also, Rotman's proofs are usually much cleaner and the overall style is very nice. It's more pleasant to read than Dummit and Foote. But this comes at a cost: Dummit and Foote do cover more material, and generalize at an earlier stage, than Rotman does.
But my biggest gripe concerns the exercises. Put simply, Rotman's are far too easy for what is being pitched as a graduate course. In fact, they are in general far easier than the homework problems I sweated through when I took honors undergraduate algebra. They're barely adequate to convince the reader that he has a basic grasp on the material, and there are almost no hard ones, let alone really tough, thought-provoking open-ended problems like one encounters in Herstein's "Topics in Algebra" (an undergraduate book). There are certainly no exercises in Rotman's book that would be of any use for a graduate student preparing for qualifying exams. They're not even much of a workout for a decent (honors student) undergraduate.
So, what is this book good for? I think it's great for reading material that is usually harder to understand elsewhere. Rotman has a real knack for clear mathematical exposition, and some of the chapters are a real joy to read. (Side note: there are also a lot of typos, at least in the first printing. The author maintains an errata list at his web site, and a second printing is coming soon. There are still many errata that he didn't catch, but they're fairly minor and do not detract significantly from the reading.) But this is simply not suitable for a primary graduate text or reference. Most good schools are going to demand more of their graduate students, and one is inevitably going to have to read Lang or Hungerford (and work through their exercises) to achieve competence at the graduate level. Rotman's book is a kinder, gentler book upon which to fall back when those books are inscrutable, as is all too common. I do recommend it highly for that purpose -- I think it's a very good secondary book.
An excellent Text.......2003-07-19
To begin with, don't let the title scare you. After having read through Rotman's book I am suprised that this text had not crossed my path earlier. It is a wonderful book and must have for any inspiring Algebraist. Moreover, I am quite shocked that the larger universities have not adopted this book.
(a) This book could quite easily be used as the standard third/fourth year undergraduate introduction to Abstract Algebra. In particular, the first four chapters provide a solid foundation for a moderate paced one semester course at which point the instructor has many different options for additional topics based on the performance of his/her class.
(b) Those students that move on to the graduate level, and obviously to a university using this book, would both be familiar with the temperment and flow of the author as well as devoid of the requirement of having to purchase another expensive Mathematics text. For example, my undergraduate Algebra text was Hungerford's and post completion the logical step, being familiar with his style, was to purchase Hungerford's graduate text. For those not familiar, let me tell you there is a night and day difference with repsect to how the material is presented.
(c) The remaining 7 chapters take the willing student on a pleasant tour of ring/module theory, some advanced group theory (for the inspiring group theorist I highly recommend the authors graduate text "Group Theory"), algebras(linear included), Homology(some cohomology) and finally some algebraic number theoretic concept under the heading of Commutative Rings III.
(d) Lastly, Rotamn does not get needlessly bogged down in any one section of the book. The flow is smooth, to the point with precise definitions, examples, and ample exercises.
I have only two negative remarks: one, the failure to include more aspects of field/Galois theory. This may be due to the author already having published a book entitled "Galois Theory". Two, the failure to devote an entire section to Finite Fileds and possibly some its applications. But this failure is minimal since, at present, the majority of Algebra texts, fail to adequately introduce and motivate Finite Fields.
Great Book!..........2002-10-07
I previously purchased Rotman's First Course in Abstract Algebra, and fell in love with it. So when I saw he a Second Abstract Algebra book, I had to have it. I am currently taking a Graduate Level Modern Algebra course, and I find this book to be a great help in my Studies. I wouldn't be as interested in Modern Algebra as I am now if it weren't for this book. I love this book and I would reccomend it to anyone who is interested in Modern Algebra, or taking a course in Modern or Abstract Algebra.
Average customer rating:
- A Classic Text
- Comprehensive but Dated
- Outdated notation and very dull
- Good introduction to groups
|
Schaum's Outline of Group Theory
B. Baumslag , and
B. Chandler
Manufacturer: McGraw-Hill
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Schaum's Outline of Modern Abstract Algebra (Schaum's)
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Schaum's Outline of General Topology
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Schaum's Outline of Abstract Algebra (Schaum's Outlines)
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Schaum's Outline of Complex Variables
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Schaum's Outline of Set Theory and Related Topics
ASIN: 0070041245 |
Book Description
The theory of abstract groups comes into play in an astounding number of seemingly unconnected areas like crystallography and quantum mechanics, geometry and topology, analysis and algebra, physics, chemistry and even biology. Readers need only know high school mathematics, much of which is reviewed here, to grasp this important subject. Hundreds of problems with detailed solutions illustrate the text, making important points easy to understand and remember.
Customer Reviews:
A Classic Text.......2006-06-22
I had been studying group theory on my own independently using this book for the last seven months when I misplaced it while at work - along with the notebook which I had painstakingly and carefully created from my hundreds of hours of study in this book. To say the least, I've been absolutely devastated at losing my notes; but the Schaum Outline I can easily replace.
I had thought, after the book was lost, of trying another text. But most of the introductory textbooks on abstract algebra cover a lot of other things besides group theory. And as a result, they do not go very deeply into any one algebraic structure, but just scratch the surface. I wanted to focus on groups because, as stated in the Introduction of this book, this will bring me into the advanced areas of more quickly as a result of the narrowness of focus.
The notation in this book is initially peculiar. I was not used to seeing the notation xf for a function instead of f(x). The lack of parentheses was confusing, so when making my notes I simply added them, creating the notation (x)f. In fact this backward notation does seem to work better for abstract algebra, and after a while it becomes natural, and the standard notation f(x) becomes odd. So expect to see such things as this for automorphisms: (a*b)f = (a)f*(b)f.
Initially I did not want to use a Schaum outline to study groups. I wanted a hard-cover textbook. But I found this book irresistible in both its scope and its detailed discussion of group concepts. In summary, I would say that if you are truly serious about abstract algebra, you cannot afford to be without this fascinating, thorough, and inexpensive text.
Comprehensive but Dated.......2005-09-13
This is a reprint of a book that's been around since the sixties. It needs an update, especially the exercises, which are somewhat disorganized. The authors, like many mathematicians, have difficulty with the spoken language and do not adequately motivate the material, on an historical or intellectual basis. That said, this is still one of the best introductions to the subject available, at less than 20% the going cost of a textbook.
Outdated notation and very dull.......2004-06-04
I know maths books aren't meant to be fun to read, but this book is *extremely* boring. It's got, in my opinion, too much content, and its content could've been explained more efficiently.
Most of the notation used in this book (it was published 36 years ago) is out of date, which can be annoying as it makes the confusion subject of group theory even more confusing.
The good thing about this book is that it's great value for money. However, as said above, it might contain too much if you're an undergraduate student like myself who just wants to understand the basic stuff.
Good introduction to groups.......2000-06-05
This book is well organized and broad for a problem-solver, and has several useful features for beginners such as classification of groups up to order 15 and complete multiplication tables for A4 and S4 (no one would take the time to actually write and print these out, but they did in this book).
I also find the problems very well-selected and are frequently used later on, so you feel you didn't just go randomly solving problems.
The authors give many examples of groups and
groupoids, ranging from isometries to Moebius transformations, and a bit of free groups and group presentations are also covered.
The Sylow Theorems are proved in the usual way, as well as the Cauchy Theorem for abelian groups, even though it is not explicitly called by that name.
Book Description
This is an introduction to Lie algebras and their applications in physics. The first three chapters show how Lie algebras arise naturally from symmetries of physical systems and illustrate through examples much of their general structure. Chapters 4 to 13 give a detailed introduction to Lie algebras and their representations, covering the Cartan-Weyl basis, simple and affine Lie algebras, real forms and Lie groups, the Weyl group, automorphisms, loop algebras and highest weight representations. Chapters 14 to 22 cover specific further topics, such as Verma modules, Casimirs, tensor products and Clebsch-Gordan coefficients, invariant tensors, subalgebras and branching rules, Young tableaux, spinors, Clifford algebras and supersymmetry, representations on function spaces, and Hopf algebras and representation rings. A detailed reference list is provided, and many exercises and examples throughout the book illustrate the use of Lie algebras in real physical problems. The text is written at a level accessible to graduate students, but will also provide a comprehensive reference for researchers.
Customer Reviews:
Mixed feelings.......2007-09-15
Lie groups and Lie algebras permeate most parts of theoretical physics. Every student in physics should have some basic notions of the subject as it sometimes tends to have unsuspected applications.
The first three chapters of this book include exemples and motivation for the more formal aspect of the Lie theory. Those are also meant to set the notation used later throughout the book. Topics covered should be well-known from a senior undergraduate student with a good background in quantum mechanics (harmonic oscillator, the rotation group) and particle physics (mostly the "zoological" part of it : classification of particles, the eightfold way and so on).
From chapter 4 on, the Maths definitely take the most prominent part of the stage. Chapter 4 is a reminder of basic notions in algebra, as covered in an undergraduate course in algebra and classical groups.
Chapter 5, on representation, should not be a challenge to the physicist.
The core of the subject is presented in chapter 6, where the idea of the Cartan-Weyl basis is given a nice presentation. This chapter is a little bit more demanding. Some statements are not proved. However, a committed student in physics, should be able to devise proofs for him/herself.
Chapter 7 is particularly enjoyable, dealing with Dynkin diagrams and the classification of finite simple Lie algebras, and introducing infinite dimensional ones. The way Kac-Moody algebras appear, through relaxing the axioms of the Chevalley-Serre construction should be appreciated. Also, physical exemples are to the point.
However, beginning with chapter 12, the wrongs of this book become somewhat annoying. For instance, in chapter 12, the authors of this book freely speak of Verma modules, highest weight representations, while these concepts are to be introduced and properly developped in later chapters. I found this chaffing from an introductory book. From chapter 12, it seems that the reader is to gently follow and accept the statements made by the author, without encountering much proof or hint to this all.
Things come more acceptable in later chapters only, where invariant tensors and other things more familiar from a physicist with no previous acquaintance to Lie algebras, are exposed.
All in all, a good book for some parts of it but whose value could have surely been enhanced by adopting a more pedagogical presentations. Some proofs to key facts in the more "exotic subjects", would have been welcome, too. All the more, that some chapters of this book did not require much work from the authors, as it seems that they were taken from Dr. Fuchs "Affine Lie algebras".
Hopefully, welcome additions will be added to a further edition.
Beginners or readers with a casual interest in Lie algebras should better learn it from another source.
Average customer rating:
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An Invitation to C*-Algebras (Graduate Texts in Mathematics)
W. Arveson
Manufacturer: Springer
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Banach Algebra Techniques in Operator Theory (Graduate Texts in Mathematics)
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Theory of Linear Operators in Hilbert Space
ASIN: 0387901760 |
Book Description
This book is an introduction to C^*-algebras and their representations on Hilbert spaces. The presentation is as simple and concrete as possible; the book is written for a second-year graduate student who is familiar with the basic results of functional analysis, measure theory and Hilbert spaces. The author does not aim for great generality, but confines himself to the best-known and also to the most important parts of the theory and the applications. Because of the manner in which it is written, the book should be of special interest to physicists for whom it opens an important area of modern mathematics. In particular, chapter 1 can be used as a bare-bones introduction to C^*-algebras where sections 2.1 and 2.3 contain the basic structure thoery for Type 1 von Neumann algebras.
Customer Reviews:
Lasting value........2002-09-02
"Operator algebras! Oh, right! That is Arveson's book, isn't it?"--From a typical conversation I would have with a colleague from the other end of campus.-- Or, an impatient student, I meet in the elevator, wants to quickly get an idea of what it is all about,-- and have fun reading about the material, at the same time: That would be Arveson's lovely little book!-- Sure, there *are* lots of great books in the subject; but they haven't had this same kind of wide impact. The central ideas are very attractively presented: It *is* an invitation! The other books in C*-algebra theory will typically be thicker, and they might be more narrowly focused, --more for the specialists, if you like. The first edition of Arveson's book is from the seventies; but still,-- now many years later, everyone knows "An invitation", and reads it. Authors keep immitating its approach and its style: You see immitators,--authors in other specialties of math writing books entitled "An invitation to ...". But none of the immitations seem to have quite the charm of the original. You can't very well **plan to** write a charming book in math. But when one arrives, we all know it. The subject of Arveson's book started with quantum theory, Hilbert space, spectral theory, and representations of groups and algebras. And, in the half century plus, since its inception, the subject has found applications in a surprisingly wide range of other fields: geometry, K-theory, fundamental physics, symbolic dynamics, and tiling theory, to mention just a few.
Book Description
This book addresses Lie groups, Lie algebras, and representation theory. In order to keep the prerequisites to a minimum, the author restricts attention to matrix Lie groups and Lie algebras. This approach keeps the discussion concrete, allows the reader to get to the heart of the subject quickly, and covers all of the most interesting examples. The book also introduces the often-intimidating machinery of roots and the Weyl group in a gradual way, using examples and representation theory as motivation. The text is divided into two parts. The first covers Lie groups and Lie algebras and the relationship between them, along with basic representation theory. The second part covers the theory of semisimple Lie groups and Lie algebras, beginning with a detailed analysis of the representations of SU(3). The author illustrates the general theory with numerous images pertaining to Lie algebras of rank two and rank three, including images of root systems, lattices of dominant integral weights, and weight diagrams. This book is sure to become a standard textbook for graduate students in mathematics and physics with little or no prior exposure to Lie theory. Brian Hall is an Associate Professor of Mathematics at the University of Notre Dame.
Customer Reviews:
Horrible.......2007-07-22
It doesn't take a lot of intelligence to figure
out how to present lie algebras and lie groups
if you are going to take the matrix route.
Namely, you give lots of concrete examples
(requiring nothing more than calculus as
background) and then just state what the general
case is. In this book, the author uselessly drags
the uninitiated through swamps of archaic notation
(save that for the real thing) and incomplete
proofs (where invariably the hard parts are just quoted)
so that you have to wonder what in the world is the point
of committing this mess to paper. It is ironic that the
very same publisher already has better books out on exactly
the same topics. Finally, if this really were an introduction
you wouldn't have to add 'elementary' to the title - so let's
call a spade a spade and leave the spin to the politicians.
Companion book suggestion.......2007-07-10
This is an excellent book on a difficult subject.
When learning Group Theory from the viewpoint of physics, one can miss out completely on some of the important mathematical aspects.
Halls book solved that problem for me. But, I can imagine that it also works in the reverse;
If one studies Group Theory from a pure mathematical viewpoint, one can miss out on a multitude of computational techniques and some important results.
The paramount example of Halls book is the handling of the representations of the group SU(3).
To gain even more insight into that group one can use Halls book together with Quantum Mechanics: Symmetries.
There one can see "Groups, Algebras and their Representaions in Action", especially SU(3),
in numerous solved excercises and problems displaying a multitude of relevant computational techniques.
The two books begin at about the same point (groups, algebras, representations, the exponential map),
and end at about the same point (classification of the classical groups).
Halls book provides the correct mathematical setting and Greiners book the solved examples.
The two books together add up to a lot of value.
The pure math student can easily ignore the physics in Greiners book and pick up some new things in representation theory,
such as Cartans criterion for irreducibility, dimension formulas for representations, etc.
Meanwhile, the pure physics student should probably avoid trying to learn Group Theory from physics books (including Greiners).
There is a lot of confusion in the physics books as to what is what. Groups, algebras, representations and invariant subspaces are constantly mixed up.
In conclusion, one benifits from a math book, and a large collection of examples. Halls book and Greiners book work surprisingly well together.
A refreshingly clear introductory text on Lie groups.......2004-04-19
I rarely have time or feel strongly enough about a text to write a review. However, with Hall's book, I feel compelled. After struggling with the rather compact sixth chapter of Wulf Rossman's book on representations of Lie groups and algebras during a course on representation theory (the first five chapters were assumed), I turned to this one, and boy, am I ever glad I did.
The main and overriding strength of this book is the willingness of the author to guide the reader in digesting definitions and proofs. This comes in the form of numerous examples and counterexamples to point the reader in the right direction after a definition. And Hall constantly reminds readers of particular relevant terms in the course of applying them, which I found very effective in reinforcing concepts, and which allowed me to focus on the task at hand rather than spending time sifting through previous chapters, often losing sight of the main point of the argument.
Another strong point is the approach taken to introducing weights and roots of particular representations. I have found this a very difficult subject (as I guess a lot of students do) and Rossman's book was not helping much. As the previous reviewer noted, this book starts out (chapters four and five) with detailed treatments of the representations of su(2) and su(3) via the complexifications sl(2; C) and sl(3; C) and introduces roots in these contexts as pairs of simultaneous eigenvalues of the basis elements of the Cartan subalgebra. This requires only a background in linear algebra to digest and really hits home the point of these constructs in the whole scheme of things. After these examples under the belt, the reader is then able to take in the general definition of a root as a linear functional in chapter six. Representations of general semisimple Lie algebras are covered in chapter seven.
Throughout it all, Hall's style is very clear and his proofs are complete and illuminating. If you have had courses in linear and modern algebra, you should be fine with this one. Very well suited for self study. I can't recommend this book highly enough.
AT LAST, LIE GROUPS & ALGEBRAS I CAN UNDERSTAND!!.......2003-09-16
This book focuses on matrix Lie groups and Lie algebras, and their relations and representations. This makes things a bit simpler, and not much is lost, because most of the interesting Lie groups & algebras are (isomorphic to)groups & algebras of matrices.
I believe that most mathematicians are more concerned with impressing their colleagues with their subtlety and erudition than they are in making a clear, simple and comprehensible presentation. This is mitigated by the publisher's insistence that the first 10 pages be clear to a mid-level undergraduate so the book will sell. So I usually get stuck at page 10 in those books.
This book is clear (to me) at least to page 168 (as far as I have progressed). There are even appendices on finite groups and key aspects of linear algebra. After introducing the classical groups and their algebras and the exponential map relating one to the other, the author introduces representations. He then details the representations of sl(2,C) and sl(3,C) (a.k.a. the complexifications of su(2) and su(3), respectively). By going through the details on these [with their Cartan subalgebras, weights, roots, Weyl groups, etc.], the general theory that follows is more palatable than it might otherwise be. Little rigor is sacrificed (if I am qualified to judge that - probably not). A few proofs are left out, but not many.
Another virtue of this book is that there are very few mistakes. I have trouble distinguishing an author's typos from my thinkos, so this is a particularly impotant feature of this book.
I very highly recoommend this book to anyone who does not already know the subject; it would be a perfect first book on this area. This book is really written with the student in mind. As a "shade - tree" mathematician, I need all the help I can get in understanding this difficult subject. Hall has done the best job I have seen at making the theory accessible without sacrificing rigor.
Average customer rating:
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Computational Methods for Representations of Groups and Algebras (Progress in Mathematics)
Manufacturer: Birkhäuser Basel
ProductGroup: Book
Binding: Hardcover
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Book Description
This book presents material from three survey lectures and 14 additional invited lectures given at the Euroconference "Computational Methods for Representations of Groups and Algebras" held at Essen University in April 1997. The purpose of this meeting was to provide a survey of general theoretical and computational methods and recent advances in the representation theory of groups and algebras. Furthermore, new applications of the computational methods in linear algebra to the revision of the classification of finite simple sporadic groups are presented. Computational tools (including high-performance computations on supercomputers) have become increasingly important for classification problems. They are also inevitable for the construction of projective resolutions of finitely generated modules over finite-dimensional algebras and the study of group cohomology and rings of invariants.
A major part of this book is devoted to a survey of algorithms for computing special examples in the study of Grothendieck groups, quadratic forms and derived categories of finite-dimensional algebras. Open questions on Lie algebras, Bruhat orders, Coxeter groups and Kazhdan Lusztig polynomials are investigated with the aid of computer programs. The representation theory of finite groups and finite-dimensional algebras are linked by the condensation technique. The contents of this book provide an overview on the present state of the art, for graduate students and researchers in mathematics, computer science and physics.
Series: Progress in Mathematics, Vol. 173
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
Lie groups has been an increasing area of focus and rich research since the middle of the 20th century. Procesi's masterful approach to Lie groups through invariants and representations gives the reader a comprehensive treatment of the classical groups along with an extensive introduction to a wide range of topics associated with Lie groups: symmetric functions, theory of algebraic forms, Lie algebras, tensor algebra and symmetry, semisimple Lie algebras, algebraic groups, group representations, invariants, Hilbert theory, and binary forms with fields ranging from pure algebra to functional analysis.
Key to this unique exposition is the large amount of background material presented so the book is accessible to a reader with relatively modest mathematical background. Historical information, examples, exercises are all woven into the text.
Lie Groups: An Approach through Invariants and Representations will engage a broad audience, including advanced undergraduates, graduates, mathematicians in a variety of areas from pure algebra to functional analysis and mathematical physics.
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