This course-based primer provides an introduction to Lie algebras and some of their applications to the spectroscopy of molecules, atoms, nuclei and hadrons. In the first part, it concisely presents the basic concepts of Lie algebras, their representations and their invariants. The second part includes a description of how Lie algebras are used in practice in the treatment of bosonic and fermionic systems. Physical applications considered include rotations and vibrations of molecules (vibron model), collective modes in nuclei (interacting boson model), the atomic shell model, the nuclear shell model, and the quark model of hadrons. One of the key concepts in the application of Lie algebraic methods in physics, that of spectrum generating algebras and their associated dynamic symmetries, is also discussed. The book highlights a number of examples that help to illustrate the abstract algebraic definitions and includes a summary of many formulas of practical interest, such as the eigenvalues of Casimir operators, and the dimensions of the representations of all classical Lie algebras.

For this new edition, the text has been carefully revised and expanded; in particular, a new chapter has been added on the deformation and contraction of Lie algebras.

From the reviews of the first edition:

Iachello has written a pedagogical and straightforward presentation of Lie algebras [...]. 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 [...] written by one of the leading experts in the field [...] 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 [...] and leaving the embarrassing theoretical presentations aside.

(Rutwig Campoamor-Stursberg, Zentralblatt MATH, Vol. 1156, 2009)