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Title: Lie Groups and Algebras With Applications to Physics, Geometry, and Mechanics (Applied Mathematical Sciences, Vol 61) by D.H. Sattinger, O.L. Weaver ISBN: 0-387-96240-9 Publisher: Springer-Verlag Pub. Date: 01 April, 1986 Format: Hardcover Volumes: 1 List Price(USD): $74.95

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Average Customer Rating: 3 (1 review)

Rating: 3
Summary: An interesting introduction
Comment: This book is intended as a first introduction to the theory of Lie groups and Lie algebras, focused on applications in physics. In its first chapters the authors introduce the material basing on important examples like the rotation algebra or the realization of the Heisenberg Lie algbebra in terms of annihilation/creation operators. This will lead to the general theory, having in mind these important physical examples. The book is essentially divided into three parts. The first is a differential geometric chapter dealing with the usual concepts of manifolds, vector fields, integration theorems, etc, but it also provides a topic which is usually not covered by textsbooks, namely the Maurer-Cartan equations of a Lie group/algebra. This is an important alternative, both from the physical as from the mathematical point of view. The second part corresponds to the algebraic theory of complex semisimple Lie algebras, which corresponds more or less to the standard contents of a textbook. The authors also briefly comment on the real forms of complex simple Lie algebras, which is an essential ingredient for physical applications (see e.g. the kinematical Lie algebras). The third part corresponds to representation theory of complex semisimple algebras, motivated by a detailed exposition of the eightfold way of Gell-Mann and Ne'eman. The study of representations also analyzes briefly tensor product decompositions. A final chapter is devoted to the application of Lie groups to the integration of Hamiltonian systems, a topic which has become of great interest in the last years. This kind of appendix is a good introduction to more advanced expositions like the monography of Fomenko and Trofimov.
Globally, the book covers the most important topics on Lie algebras/groups that are necessary for physical applications. Many proper notations like Pauli and Gell-Mann matrices are used, and each section is completed with a set of exercises. The book presents only very few misprints, like in the tensor product of the standard representation of the su(3) algebra.
It is very recommendable as an introductory text to Lie theory.