Group Theory: Birdtracks, Lie’s, and Exceptional Groups by Predrag Cvitanović (PDF)

Description

If classical Lie groups preserve bilinear vector norms, what Lie groups preserve trilinear, quadrilinear, and higher order invariants? Answering this question from a fresh and original perspective, Predrag Cvitanovic takes the reader on the amazing, four-thousand-diagram journey through the theory of Lie groups. This book is the first to systematically develop, explain, and apply diagrammatic projection operators to construct all semi-simple Lie algebras, both classical and exceptional. The invariant tensors are presented in a somewhat unconventional, but in recent years widely used, “birdtracks” notation inspired by the Feynman diagrams of quantum field theory. Notably, invariant tensor diagrams replace algebraic reasoning in carrying out all group-theoretic computations. The diagrammatic approach is particularly effective in evaluating complicated coefficients and group weights, and revealing symmetries hidden by conventional algebraic or index notations. The book covers most topics needed in applications from this new perspective: permutations, Young projection operators, spinorial representations, Casimir operators, and Dynkin indices. Beyond this well-traveled territory, more exotic vistas open up, such as “negative dimensional” relations between various groups and their representations. The most intriguing result of classifying primitive invariants is the emergence of all exceptional Lie groups in a single family, and the attendant pattern of exceptional and classical Lie groups, the so-called Magic Triangle. Written in a lively and personable style, the book is aimed at researchers and graduate students in theoretical physics and mathematics.

User’s Reviews

Editorial Reviews: Review “This book has to be seen to be believed! The title, Group Theory, is nothing if not surprising, given that the material dealt with by Predrag Cvitanović in these roughly 250 pages requires a level of sophistication well beyond what is offered in the early stages of university algebra. In point of fact, the general theme of the book under review is Lie theory with representation theory in the foreground, and Cvitanović’s revolutionary goal (e.g., ‘birdtracks’) and, for lack of a better word, the attendant combinatorics. . . . [F]or the right reader, which is to say, an R>0-linear combination of mathematician and physicist equipped with a zeal for novel combinatorics flavored diagram-gymnastics, this book will be a treat and a thrill, and its new and radical way to compute many things Lie is bound to make its mark.”—Michael Berg, MAA Reviews”More than just an innovative notation, this book offers a conceptually novel alternative path to a key mathematical result, the classification of finite-dimensional simple Lie algebras. . . . While this volume is an obvious resource for physics students, the traces of physics that remain in the work will elucidate for mathematics students how physics uses Lie groups as a tool.”—D.V. Feldman, Choice”I think that the book is a very interesting and thought provoking contribution to the literature on representations of compact Lie groups. It has many interesting original aspects that deserve to be known much better than they are.”—Karl-Hermann Neeb, Journal of the Lie Theory”[T]he narrative of the book is written in a relaxed and witty style. The book is intriguing as well as entertaining.”—Jeb F. Willenbring, Mathematical Reviews Review “In Group Theory, we get a sense of the quest Cvitanović has been on, which makes the book much more fun to read than the average mathematics text. This book is intriguing, novel, and important.”―John Baez, University of California, Riverside”There has been an urgent need for an in-print and readily available version of Cvitanović’s innovative and systematic approach to the group-theoretic calculations occurring in theoretical physics and beyond. Well-organized and well-written, this book is definitely an important and valuable contribution to its field.”―Alan J. Macfarlane, Cambridge University From the Inside Flap “In Group Theory, we get a sense of the quest Cvitanovic has been on, which makes the book much more fun to read than the average mathematics text. This book is intriguing, novel, and important.”–John Baez, University of California, Riverside”There has been an urgent need for an in-print and readily available version of Cvitanovic’s innovative and systematic approach to the group-theoretic calculations occurring in theoretical physics and beyond. Well-organized and well-written, this book is definitely an important and valuable contribution to its field.”–Alan J. Macfarlane, Cambridge University From the Back Cover “In Group Theory, we get a sense of the quest Cvitanovic has been on, which makes the book much more fun to read than the average mathematics text. This book is intriguing, novel, and important.”–John Baez, University of California, Riverside”There has been an urgent need for an in-print and readily available version of Cvitanovic’s innovative and systematic approach to the group-theoretic calculations occurring in theoretical physics and beyond. Well-organized and well-written, this book is definitely an important and valuable contribution to its field.”–Alan J. Macfarlane, Cambridge University About the Author Predrag Cvitanović is the Glen P. Robinson Professor of Nonlinear Science at the Georgia Institute of Technology. He is the author of Universality in Chaos and lead author of the ChaosBook.org webbook. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Any practitioner of the famous Bourbaki group would have a brain implosion if they tried to read this book. Their aversion to “practical” and “visual” tools in mathematics as opposed to the rigorous formalisms for which they are justifiably famous is well documented by Benoit Mandelbrot, amongst others. They would not like this work.This book is, first and foremost, a book about beauty with beautiful ideas and striking visuals. This is deliberately so. As the author states, “if diagrammatic notation is to succeed, it need not only be precise, but also beautiful”. Of course one would expect that a book whose subject matter is primarily the many elegant symmetries of nature would end up dealing with beauty, not only at the mathematical but also at the visual level. I can testify to the success of Professor Cvitanovic’s endeavors on this front. When the book arrived I showed it to my wife, who has some artistic training and a plus-critical eye for visuals, (but who certainly can’t tell the difference between a casimir and a casserole). Her response said it all.To fully appreciate this book you need postgraduate qualifications in a slew of highly technical subjects and I won’t pretend that my qualifications are anywhere nearly sufficient for this. But I am mathematically literate enough and have enough formal training, (and enough interest in many of the topics covered), to recognize that this is a beautiful and brilliant work which will give me and other readers years of additional pleasure. I say “additional” because I have been tracking Professor Cvitanovic’s web book, from which this print originates, for a long time now and the contents of this work are not new to me.Take note that this is not a textbook or even a standard reference of any sort and it makes no concessions whatsoever to lack of prior knowledge. It is primarily a highly technical research monograph about problem solving. Each chapter launches headfirst into its subject matter without offering any but the most cursory introduction. The reader is expected to understand the nomenclature, the context of the material and be familiar with the mathematics and physics dealt with. But even with this, the amazing clarity of Professor Cvitanovic’s writing shines through. In contrast to the dense and murky scribblings of many (I might even suggest, most) texts at this level, the writing in this book as well as the flow of logic, the layout and everything else which matters is breathtakingly simple and, may I say, beautiful. The work deserves 10 stars out of 5. (I’m still working on the mathematics of that!)I only wish that the good professor would see his way clear to publishing in hard cover his other web book, ChaosBook.pdf, which is an equally elegant and valuable work and which aligns much better with material that I am very familiar with. Perhaps one day…I strongly urge all interested readers to purchase this book in hard cover, even if they have the web book version. This is not only for the considerable value of the book itself, but also as a way of showing appreciation for the remarkable generosity and altruism of the author in making the fruits of years of hard labor available for free.

⭐I bought this book because I needed a modern treatise on Group Theory and because I would like to plug in some holes in my knowledge of it. Originally I gave it 4 stars rating but after reading it for a month I am changing it to 3 stars.PROS:(1) Very nice uniform and modern approach to group theory based on projection operators.(3) Well structured presentationCONS:(1) Book is short counting only 273 pages. Applications are covered rather lightly. I wish the book was twice the size with second half devoted to gory details with applications in physics, geometry, etc.(2) Book is terse and intense. It is assumed that a reader knows Linear Algebra very well including spectral theory. Paul Halmos “Final Dimensional Vector Spaces” should help there. I guess it is also implied that a reader has been exposed to group theory already.(3) The book is not a text book, it is a monograph on the subject.(4) Diagrammatic tools used, aka “birdtracks”, are not very useful at least to me. I find myself constantly rewriting the birdtracks diagrams back into tensor notations. This is in contrast with Feynman diagrams that I use and love.Overall I gave it 3 (three) stars. It definitely gave me food for thought but I learned little new from it.

⭐This is not my field, honestly I bought it on impulse because find the main concept too cute.

⭐Beautiful exposition for an obscure subject. You get a good summary of what group theory is all about whilst learning diagrammatics.

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