Group theory : birdtracks, Lie's, and exceptional groups
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Bibliographic Information
Group theory : birdtracks, Lie's, and exceptional groups
Princeton University Press, c2008
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Includes bibliographical references and index
Description and Table of Contents
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.
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
by "Nielsen BookData"