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Lie Groups, Lie Algebras, and Representations: An Elementary Introduction (Graduate Texts in Mathematics) [Englisch] [Gebundene Ausgabe]

Brian Hall
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Kurzbeschreibung

27. August 2004 0387401229 978-0387401225 1st ed. 2003. Corr. 2nd printing 2004

Lie groups, Lie algebras, and representation theory are the main focus of this text. 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.


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From the reviews:

"This is an excellent book. It deserves to, and undoubtedly will, become the standard text for early graduate courses in Lie group theory … . It is clearly written … . A reader of this book will be rewarded with an excellent understanding of Lie groups … . Hall’s book appears to be genuinely unique in both the organization of the material and the care in which it is presented. This is an important addition to the textbook literature … . It is highly recommended." (Mark Hunacek, The Mathematical Gazette, March, 2005)

"The book is written in a systematic and clear way, each chapter ends with a set of exercises. The book could be valuable for students of mathematics and physics as well as for teachers, for the preparation of courses. It is a nice addition to the existing literature." (EMS-European Mathematical Society Newsletter, September, 2004)

"This book differs from most of the texts on Lie Groups in one significant aspect. … it develops the whole theory on matrix Lie groups. This approach … will be appreciated by those who find differential geometry difficult to understand. … each of the eight chapters plus appendix A contain a good collection of exercises. … I believe that the book fills the gap between the numerous popular books on Lie groups … is a valuable addition to the collection of any mathematician or physicist interested in the subject." (P.K. Smrz, The Australian Mathematical Society Gazette, Vol. 31 (2), 2004)

"This book addresses Lie groups, Lie algebras, and representation theory. … 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 the most interesting examples. … 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." (L’Enseignement Mathematique, Vol. 49 (3-4), 2003)

"Though there exist already several excellent text books providing the mathematical basis for all this, introductions aimed at graduate students both in mathematics and physics seem to be rare. So the guiding principle in the planning of the book by Brian Hall … was to minimize the amount of prerequisites. … students will benefit from the way the material is presented in this Introduction; for it is elementary and not intimidating, at the same time very systematic, rigorous and modern … ." (G. Roepstorff, Zentralblatt MATH, Vol. 1026, 2004)

"This book is a great find for those who want to learn about Lie groups or Lie algebras and basics of their representation theory. It is a well-written text which introduces all the basic notions of the theory with many examples and several colored illustrations. The author … provides many informal explanations, several examples and counterexamples to definitions, discussions and warnings about different conventions, and so on. … It would also make a great read for mathematicians who want to learn about the subject." (Gizem Karaali, MAA Mathematical Sciences Digital Library, January, 2005)

"Lie groups are already standard part of graduate mathematics, but their complex nature makes still a challenge to write a good introductory book to it. … This book is a must for graduate students in mathematics and/or physics." (Árpád Kurusa, Acta Scientiarum Mathematicarum, Vol. 73, 2007)

“The book under review therefore makes the wise choice of sticking to linear groups. … Hall’s book has two parts. In the first part, ‘General theory’, the author introduces matrix Lie groups … . A highlight of the second part is the discussion of 3 different constructions of irreducible representations of complex semisimple Lie algebras: algebraic (Verma modules), analytic (Weyl character formula), geometric (Borel-Weil construction using the complex structure on the flag manifold). … this book is a fine addition to the literature … .” (Alain Valette, Bulletin of the Belgian Mathematical Society, 2009)

Synopsis

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.


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2 von 2 Kunden fanden die folgende Rezension hilfreich
Format:Gebundene Ausgabe
Für Studenten, die eine moderne, mathematisch absolut rigorose aber trotzdem ohne allzu große mathematische Vorkenntnisse lesbare Einführung in die Theorie von Matrix-Lie-Gruppen, Lie-Algebren und der Darstellungstheorie suchen, ist dieses Buch optimal geeignet: Durch die Konzentration auf Matrix-Lie-Gruppen ist es dem Autor möglich, die Theorie ohne Vorkenntnisse aus der Differentialgeometrie zu entwickeln. Gebraucht werden nur Grundbegriffe aus der Topologie. Für einige kniffligere Begriffe und Beweise reicht ein elementares Verständnis der Homotopietheorie und der Überlagerungstheorie. Wenn man diese Begriffe nicht kennt, ist das Buch aber genausogut lesbar, da die Begriffe wirklich nur selten verwendet werden. Damit eröffnet das Buch insbesondere Physik-Studenten, die ein solides Verständnis der Darstellungstheorie gewinnen wollen, einen Zugang zu diesem komplexen Thema. Andere Bücher zu diesem Thema sind entweder ziemlich oberflächlich (typischerweise "Physik"-Bücher über Darstellungstheorie) oder überfordern die meisten Studenten, da ihre Mathematik-Kenntnisse einfach nicht umfassend genug sind. Allen Physikern, denen die "handwaving" - Erklärungen der meisten Physik-Bücher einfach nicht reichen sei dieses Buch wärmstens empfohlen: Ich denke für den Einstieg wird man kein besseres Buch auf dem Markt finden. Für Mathematiker, die in das Thema einsteigen wollen, ist dieses Buch aber genauso geeignet: Wenn man das Buch von Hall verstanden hat kann man danach immer noch zu den anspruchsvolleren Titeln übergehen. Was man bei Hall gelernt hat wird einem hier mit Sicherheit zugute kommen.
War diese Rezension für Sie hilfreich?
Die hilfreichsten Kundenrezensionen auf Amazon.com (beta)
Amazon.com: 3.9 von 5 Sternen  9 Rezensionen
99 von 102 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen AT LAST, LIE GROUPS & ALGEBRAS I CAN UNDERSTAND!! 16. September 2003
Von Reed B Wickner - Veröffentlicht auf Amazon.com
Format:Gebundene Ausgabe
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.
22 von 25 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Companion book suggestion 10. Juli 2007
Von Johan Nystrom - Veröffentlicht auf Amazon.com
Format:Gebundene Ausgabe
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, derivations of 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.
4 von 4 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Excellent introduction into the theory of Lie Groups 23. Februar 2008
Von Richard - Veröffentlicht auf Amazon.com
Format:Gebundene Ausgabe
Brian Hall's book is a welcome addition to the material available for the study of Lie Groups. This book in particular provides a good basis for the study of Lie Groups without getting caught up in the study of Manifold Theory. The book is easy to access, requiring only a basic background in Modern and Linear Algebra and has many applications pertaining both to mathematics and physics.
3 von 3 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Best book out there to learn Lie theory from 15. Juni 2013
Von Colim - Veröffentlicht auf Amazon.com
Format:Gebundene Ausgabe|Verifizierter Kauf
I used this text last year for a third year undergraduate course on Lie groups, Lie algebras and representations and it was excellent! I found the exposition to be super clear and easy to follow. Also, I really loved an entire chapter that focused on the representation theory of $\mathfrak{sl}_3(\Bbb{C})$! As I grow older, I discover more and more the importance of having a concrete grasp of examples - and Brian's book does it in exactly that way. Plus, it is not disorganized and does not handwave like Fulton and Harris (which has gaps in its proofs using Weight diagrams in the chapter on representations of $\mathfrak{sl}_3$) and is not dry and abstract like Humphreys' book!

I would say that this is the best book out there to start learning Lie theory from.
3 von 3 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen This is a great book 9. Juni 2013
Von Greg - Veröffentlicht auf Amazon.com
Format:Gebundene Ausgabe|Verifizierter Kauf
I love this book. I like to think I am a good mathematician, but I have always had a lot of trouble with differential geometry. I suspect there are a lot of people out there like me. This book presents Lie Groups using matrix groups, which makes things much more concrete. The book is not easy, and requires good linear algebra skills. However, many matrix algebra theorems are presented and proved in the appendices. The appendices also include the abstract definitions of Lie groups and algebras for general manifolds which are topological groups, with examples, and the author always explains how the theorems for matrix groups relate to those for general Lie groups, and in many cases little modification seems to be necessary.

There are plenty of exercises at the end of each chapter that are of just the right difficulty to help you understand the material.

I got the first edition, since the second edition seems to be available only in paperback and I prefer hardcover. There seem to be a lot of typos in the first edition that are easy to spot. I imagine they have been corrected in the second edition. I don't know what other changes were made.

Some people who reviewed this book complained that it does not fully reveal the pristine beauty of general Lie groups. This may be true, but a book that I cannot read is not going to do me any good. Also, many people who use Lie Groups only really need matrix groups and this may be a good book for them.
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