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Groups and Symmetries

Yvette Kosmann-Schwarzbach
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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This is the second edition of a book that was first published in 2009. For background, please see our review of the first edition. The author describes this edition as a “revised and expanded” version of the first. The two most visible changes, and the only two specifically mentioned by the author, are the inclusion of historical sections at the end of each chapter (which are generally excellent, and add considerably to the book), and the addition of a new chapter on spin groups and spinors (which also discusses Clifford algebras).  One very good feature of this book that has not changed is the annotated bibliography that appears at the end of each chapter and contains helpful comments about other books. 
The prerequisites for the book are somewhat unclear. The author takes the trouble to define (albeit quickly) such basic terms as “group” and “subgroup”, yet in section 5.2 refers to the “space of sections of a vector bundle” with no prior definition of that term. The idea of integrating with respect to a measure is mentioned in section 3.2, again without prior definition. 
The additions to the second edition add only about 50 pages to the (very concise) actual text, so this new edition is still quite short. The author achieves this brevity by pretty much dispensing with much in the way of expository, motivational discussion. (“Bones with no flesh”, is how a friend of mine describes the exposition.) Given this, I can’t help but feel that the back-cover blurb for this book, which says it is “[a]ccessible to advanced undergraduates in mathematics and physics as well as beginning graduate students” is overly optimistic; maybe this book can be used at a school like the École Polytechnique (the author’s institution), but I can’t see it being used as an undergraduate text at an average American university. Even for beginning graduate students, I don’t see this as a good choice for a text (unless accompanied by excellent lectures filling in details and motivation) or for self-study; the book is so concise and lacking in “flesh” that most beginning students, even with some mathematical sophistication, might find themselves looking elsewhere to fill in the gaps. 
Where might they look? For a more leisurely and chatty (but considerably less ambitious) introduction to matrix Lie groups, an undergraduate reader might prefer Stillwell’s Naïve Lie Theory, or Tapp’s Matrix Groups for Undergraduates.  Michael Artin’s book Algebra also contains several chapters discussing some of the material in the book under review. Beginning graduate students might benefit from Hall’s Lie Groups, Lie Algebras and Representations.  None of these books, however, have the focus on physics that this book does. A book that does focus on physics, and has substantial overlap with this one, is Sternberg’s Group Theory and Physics, but that book is about twice as long as this one. Another book, one that perhaps caters more to an audience of physicists than mathematicians, is Georgi’s Lie Algebras in Particle Physics: From Isospin to Unified Theories, which also has the advantage of being open access.
This book’s “highest and best use”, to borrow a real estate term, is, I think, for people who already have a pretty good understanding of Lie theory, but who want a source where they can get information, especially about physics, in a hurry. Such people might enjoy the succinct approach to the subject presented here in a way that people who are trying to learn this material for the first time would not.
Mark Hunacek ([email protected]) is a Teaching Professor Emeritus at Iowa State University.