Group Theory


Table of Contents  Introduction [PDF] 

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, fourthousanddiagram journey through the theory of Lie groups. This book is the first to systematically develop, explain, and apply diagrammatic projection operators to construct all semisimple 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 grouptheoretic 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 welltraveled 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 socalled Magic Triangle. Written in a lively and personable style, the book is aimed at researchers and graduate students in theoretical physics and mathematics. 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. Endorsements: "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 inprint and readily available version of Cvitanovic's innovative and systematic approach to the grouptheoretic calculations occurring in theoretical physics and beyond. Wellorganized and wellwritten, this book is definitely an important and valuable contribution to its field."Alan J. Macfarlane, Cambridge University. Subject Areas: 
svn: $Author: predrag $  $Date: 20081122 22:05:42 0500 (Sat, 22 Nov 2008) $