Group theory : birdtracks, Lie's, and exceptional groups / Predrag Cvitanović.

Format:

Book

Author/Creator:

Cvitanović, Predrag.

Language:

English

Subjects (All):

Group theory.

Physical Description:

1 online resource (278 p.)

Edition:

Course Book

Place of Publication:

Princeton, N.J. : Princeton University Press, c2008.

Language Note:

English

Summary:

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.

Contents:

Frontmatter

Contents

Acknowledgments

Chapter One. Introduction

Chapter Two. A preview

Chapter Three. Invariants and reducibility

Chapter Four. Diagrammatic notation

Chapter Five. Recouplings

Chapter Six. Permutations

Chapter Seven. Casimir operators

Chapter Eight. Group integrals

Chapter Nine. Unitary groups

Chapter Ten. Orthogonal groups

Chapter Eleven. Spinors

Chapter Twelve. Symplectic groups

Chapter Thirteen. Negative dimensions

Chapter Fourteen. Spinors' symplectic sisters

Chapter Fifteen. SU(n) family of invariance groups

Chapter Sixteen. G2 family of invariance groups

Chapter Seventeen. E8 family of invariance groups

Chapter Eighteen. E6 family of invariance groups

Chapter Nineteen. F4 family of invariance groups

Chapter Twenty. E7 family and its negative-dimensional cousins

Chapter Twenty-One. Exceptional magic

Appendix A. Recursive decomposition

Appendix B. Properties of Young projections

Bibliography

Index

Notes:

Description based upon print version of record.

Includes bibliographical references (p. [251]-268) and index.

ISBN:

9786612964800

9781282964808

1282964801

9781400837670

1400837677

OCLC:

705945735