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Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, by Brian Hall
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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.
- Published on: 2015-05-30
- Dimensions: 9.21" h x .78" w x 6.14" l,
- Binding: Paperback
- 376 pages
Review
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." (�rp�d 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)
Most helpful customer reviews
106 of 110 people found the following review helpful.
AT LAST, LIE GROUPS & ALGEBRAS I CAN UNDERSTAND!!
By Reed B Wickner
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.
23 of 27 people found the following review helpful.
Companion book suggestion
By Johan Nystrom
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.
6 of 6 people found the following review helpful.
Excellent introduction into the theory of Lie Groups
By Richard
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.
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