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Group Theory: Birdtracks, Lie's, and Exceptional Groups

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Overview

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.

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Editorial Reviews

Mathematical Reviews
[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
MAA Reviews - Michael Berg
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 Cvitanovic 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 Cvitanovic'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.
Choice - D.V. Feldman
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.
Journal of the Lie Theory - Karl-Hermann Neeb
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.
Mathematical Reviews - Jeb F. Willenbring
[T]he narrative of the book is written in a relaxed and witty style. The book is intriguing as well as entertaining.
From the Publisher
"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 Cvitanovi? 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 Cvitanovi?'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

MAA Reviews
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 Cvitanovic 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 Cvitanovic'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
Choice
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
Journal of the Lie Theory
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
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Product Details

  • ISBN-13: 9780691118369
  • Publisher: Princeton University Press
  • Publication date: 7/1/2008
  • Pages: 280
  • Product dimensions: 6.10 (w) x 9.30 (h) x 0.90 (d)

Meet the Author

Predrag Cvitanovic 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.

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Table of Contents


Acknowledgments xi
Chapter 1: Introduction 1
Chapter 2: A preview 5
Chapter 3: Invariants and reducibility 14
Chapter 4: Diagrammatic notation 27
Chapter 5: Recouplings 42
Chapter 6: Permutations 49
Chapter 7: Casimir operators 60
Chapter 8: Group integrals 76
Chapter 9: Unitary groups 82
Chapter 10: Orthogonal groups 118
Chapter 11: Spinors 132
Chapter 12: Symplectic groups 148
Chapter 13: Negative dimensions 151
Chapter 14: Spinors’ symplectic sisters 155
Chapter 15: SU(n) family of invariance groups 162
Chapter 16: G2 family of invariance groups 170
Chapter 17: E8 family of invariance groups 180
Chapter 18: E6 family of invariance groups 190
Chapter 19: F4 family of invariance groups 210
Chapter 20: E7 family and its negative-dimensional cousins 218
Chapter 21: Exceptional magic 229
Epilogue 235
Appendix A.Recursive decomposition 237
Appendix B.Properties of Young projections 239
H. Elvang and P. Cvitanovi´c
B.1 Uniqueness of Young projection operators 239
B.2 Orthogonality 240
B.3 Normalization and completeness 240
B.4 Dimension formula 241
Bibliography 243
Index 259
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