# Lie Groups and Lie Algebras: A Physicist's Perspective

## Adam M. Bincer

### Abstract

This book is based on lectures given to graduate students in physics at the University of Wisconsin-Madison. Group theory has been around for many years and the only thing new in this book is my approach to the subject, in particular the attempt to emphasize its beauty. The inspiration was this quote from Hermann Weyl: “My work always tried to unite the true with the beautiful; but when I had to choose one or the other I usually chose the beautiful”. The book starts with the definition of basic concepts such as group, vector space, algebra, Lie group, Lie algebra, simple and semisimple groups, ... More

This book is based on lectures given to graduate students in physics at the University of Wisconsin-Madison. Group theory has been around for many years and the only thing new in this book is my approach to the subject, in particular the attempt to emphasize its beauty. The inspiration was this quote from Hermann Weyl: “My work always tried to unite the true with the beautiful; but when I had to choose one or the other I usually chose the beautiful”. The book starts with the definition of basic concepts such as group, vector space, algebra, Lie group, Lie algebra, simple and semisimple groups, compact and non-compact groups. Next SO(3) and SU(2) are introduced as examples of elementary Lie groups and their relation to Physics and angular momentum. All irreducible representations, addition of angular momentum, Clebsch–Gordan coefficients and the Wigner–Eckart theorem are presented. Next the so(n) algebras and spinors are discussed using Clifford numbers. Conjugate, orthogonal and symplectic representations of spinors are described and the Clebsch–Gordan series for spinors is given, followed by a description of the center and outer automorphisms of Spin(n). The book next presents a diversion on Hurwitz’s theorem, quaternions and octonions, which leads into a discussion of the exceptional group G 2. The discussion of orthogonal groups is concluded with a presentation of their Casimir operators. As a lead in to unitary groups the book discusses classical groups and obtain the dimensions of orthogonal, unitary and symplectic groups in one fell swoop by treating them as unitary groups over the reals, the complex and the quaternions, respectively. The symmetric group Sr is introduced and used to discuss irreducible representations of SU(n). The next three chapters involve the Cartan basis, the Cartan classification of semisimple algebras and Dynkin diagrams. The last three chapters are of particular importance to physicists as they describe the space-time groups known as the Lorentz, Poincaré and Liouville groups and the energy levels of the hydrogen atom in n space dimensions. At the end of each chapter brief biographical notes are given on the scientist(s) mentioned in that chapter for the first time.

*Keywords: *
Lie groups,
Lie algebras,
Clebsch–Gordan coefficients,
Wigner–Eckart theorem,
Spinors,
Hurwitz’s theorem,
quaternions,
octonions,
Cartan classification,
Dynkin diagrams,
Lorentz group,
Poincaré group,
Liouville group,
hydrogen atom

### Bibliographic Information

Print publication date: 2012 |
Print ISBN-13: 9780199662920 |

Published to Oxford Scholarship Online: January 2013 |
DOI:10.1093/acprof:oso/9780199662920.001.0001 |