Lie Algebras and Applications

1. Front Matter

   Pages I-XIV

   PDF

2. Basic Concepts

   • Francesco Iachello

   Pages 1-13

3. Semisimple Lie Algebras

   • Francesco Iachello

   Pages 15-25

4. Lie Groups

   • Francesco Iachello

   Pages 27-38

5. Irreducible Bases (Representations)

   • Francesco Iachello

   Pages 39-62

6. Casimir Operators and Their Eigenvalues

   • Francesco Iachello

   Pages 63-74

7. Tensor Operators

   • Francesco Iachello

   Pages 75-89

8. Boson Realizations

   • Francesco Iachello

   Pages 91-130

9. Fermion Realizations

   • Francesco Iachello

   Pages 131-146

10. Differential Realizations

    • Francesco Iachello

    Pages 147-153

11. Matrix Realizations

    • Francesco Iachello

    Pages 155-162

12. Spectrum Generating Algebras and Dynamic Symmetries

    • Francesco Iachello

    Pages 163-172

13. Degeneracy Algebras and Dynamical Algebras

    • Francesco Iachello

    Pages 173-188

14. Back Matter

    Pages 179-190

    PDF

In the second part of the 20th century, algebraic methods have emerged as a powerful tool to study theories of physical phenomena, especially those of quantal systems. The framework of Lie algebras, initially introduced by - phus Lie in the last part of the 19th century, has been considerably expanded to include graded Lie algebras, in?nite-dimensional Lie algebras, and other algebraic constructions. Algebras that were originally introduced to describe certainpropertiesofaphysicalsystem,inparticularbehaviorunderrotations and translations, have now taken center stage in the construction of physical theories. This book contains a set of notes from lectures given at Yale Univ- sity and other universities and laboratories in the last 20 years. The notes are intended to provide an introduction to Lie algebras at the level of a one-semester graduate course in physics. Lie algebras have been particularly useful in spectroscopy, where they were introduced by Eugene Wigner and Giulio Racah. Racah’s lectures were given at Princeton University in 1951 (Group Theory and Spectroscopy) and they provided the impetus for the initial applications in atomic and nuclear physics. In the intervening years, many other applications have been made. This book contains a brief account of some of these applications to the ?elds of molecular, atomic, nuclear, and particle physics. The application of Lie algebraic methods in Physics is so wide that often students are overwhelmed by the sheer amount of material to absorb.

• Abstract algebra
• Boson
• Eigenvalue
• Matrix
• Quark
• hadron
• molecule

From the reviews:

"Iachello has written a pedagogical and straightforward presentation of Lie algebras and some applications to bosonic systems encountered in molecular, atomic, nuclear and particle physics. … The book should be of interest to graduate students and researchers in physics, although mathematicians and chemists should find it useful as well. It is a great text to accompany a course on Lie algebras and their physical applications." (Marc de Montigny, Mathematical Reviews, Issue, 2007 i)

"This book … a practical introduction to important facts concerning Lie algebras that continuously appear in physical problems, and written by one of the leading experts in the field. … the book will certainly be of great use for students or specialists that want to refresh their knowledge on Lie algebras applied to physics. … an excellent reference for those interested in acquiring practical experience in the application and techniques of Lie algebras to physics, and leaving the embarrassing theoretical presentations aside." (Rutwig Campoamor-Stursberg, Zentralblatt MATH, Vol. 1156, 2009)

• Department of Physics, Yale University, New Haven, USA

  Francesco Iachello

Policies and ethics