Lie Groups, Lie Algebras, and Some of Their Applications

Description

the original, unabridged version of Gilmore's book

This book is a reprint of the original edition of 1974 which appeared under the same title at J. Wiley. It presents the theory of Lie groups and algebras in a mathematical physicist's fashion. The author probably thinks of readers who have almost no knowledge of modern mathematical concepts but who are able to follow complicated computations by their intuition. Consequently, the method of the presentation is not rigorous mathematical proof but illustration through well-chosen examples, heuristic reasoning and applications to physics. For a mathematician this book might be helpful in providing a lot of worked out examples which usually are left as exercises for the reader in many books devoted to mathematical rigor (e.g. computations necessary for the classification of the real forms of the complex simple Lie algebras).
In the first chapter the author tries to introduce groups, fields, vector spaces and algebras, and even sets. In reading the book, one could skip this chapter without any loss. Either one knows the concepts introduced there or one has little chance to learn them from the text. The second chapter gives an informative survey on classical groups; the multilinear algebra used there relies heavily on vector space bases. In the third chapter we find conventional definitions of topological groups and Lie groups and some of their properties (like the existence of the left-invariant measure). In Chapter 4 the author discusses the relations between Lie groups and their Lie algebras. Lie's fundamental theorems are proved in Lie's spirit using structural constants and infinitesimal generators. Chapter 5 is more original; it contains remarkable examples and applications to physics. Among the topics discussed there one finds the computation of all unitary representations of the coverings of SU(2) and their non-canonical representations together with physical consequences. Chapter 6 is devoted to the classical complex and real simple Lie groups and their Lie algebras. Standard bases and commutator relations between their elements are computed and embeddings of real into complex Lie groups are studied. Various canonical decompositions of classical Lie algebras corresponding to homogeneous spaces are computed explicitly. Moreover, the author begins the study of Riemannian symmetric spaces as coset spaces of Lie groups. In Chapter 7 the structure theory of Lie algebras is treated. For solvable algebras Lie's theorem on the existence of a simultaneous eigenvector is shown. Furthermore we find criteria for solvability. The complete reducibility of semisimple Lie algebras is presented and the commutator relations are computed. The Cartan-Killing form is used to embed the root systems in Euclidean spaces. In Chapter 8 a list of all simple complex Lie algebras is given using their root spaces and their Dynkin diagrams. In Chapter 9 the author first develops the algebraic machinery necessary for the classification of the real forms of complex simple Lie algebras. Then he describes all real simple classical Lie algebras in detail. The exceptional Lie algebras are merely listed. With respect to them the author writes: "… the calculations of them seem to be of academic interest to the physicists''. The second part of this chapter contains a study of symmetric spaces as coset spaces of classical Lie groups. A formal highlight of this treatment is the list of all real forms of (pseudo-) Riemannian globally symmetric coset spaces. To do this the author needs analytic properties and representations of symmetric spaces. The presentation of these facts is very concrete and in my opinion is one of the most charming parts of the book. I like especially the various parametrizations of the Hermitian symmetric spaces given there. In the last chapter the author discusses non-semisimple Lie algebras called contractions. They arise in the following way: Consider some continuous family Uε (ε>0) of coordinate transformations of a simple Lie algebra L which has a singular linear transformation U0 as limit. For a given basis of L consider the limit of the structure constants with respect to the images of this basis under the family of mappings Uε for ε→0. This gives structure constants of a Lie algebra L0 on the vector space L, and this Lie algebra L0 is a contraction of L. The author shows that such contractions exist relative to any Cartan decomposition of a semisimple Lie algebra. He is also interested in the characterization of those non-semisimple Lie algebras which can be obtained as contractions. The enclosed bibliography on Lie groups and symmetric spaces offers a balanced view of this area of mathematics up to the year 1972 from the theoretical point of view as well as from the point of view of mathematical physics.

Reviewed by K. Strambach