Theory of Lie Groups

by Anthony E. Clement (Author),‎ Stephen Majewicz (Author),‎ Marcos Zyman (Author)

This is the second of three major volumes which present a comprehensive treatment of the theory of the main classes of special functions from the point of view of the theory of group representations. This volume deals with the properties of special functions and orthogonal polynomials (Legendre, Gegenbauer, Jacobi, Laguerre, Bessel and others) which are related to the class 1 representations of various groups. The tree method for the construction of bases for representation spaces is given. `Continuous' bases in the spaces of functions on hyperboloids and cones and corresponding Poisson kernels are found. Also considered are the properties of the q-analogs of classical orthogonal polynomials, related to representations of the Chevalley groups and of special functions connected with fields of p-adic numbers. Much of the material included appears in book form for the first time and many of the topics are presented in a novel way.

Following the publication of the book "Special Functions and the Theory of Group Representations" in 1965 by one of the authors of the present book, interest in the investigations of the different relations between these branches of mathematics, at first sight far removed from each other, has considerably increased. Great interest in this subject has been shown by physicists - almost every issue of the "Journal of Mathematical Physics" contains papers on this topic. At the same time, physical problems stimulated the study of Clebsch- Gordan coefficients, Racah coefficients, 3mj symbols, and other objects which turned out to be special functions having discrete argument. A series of impor- tant results on this topic is contained in the book "Matrix Elements and Clebsch-Gordan Coefficients of Group Representations", Kiev, 1979, by the second author of the present work. We also mention the books by W. Miller [30, 31] and J. Talman [45], devoted to related problems.

This book collects a series of contributions addressing the various contexts in which the theory of Lie groups is applied. A preliminary chapter serves the reader both as a basic reference source and as an ongoing thread that runs through the subsequent chapters.

This book is devoted to explaining a wide range of applications of con­ tinuous symmetry groups to physically important systems of differential equations. Emphasis is placed on significant applications of group-theoretic methods, organized so that the applied reader can readily learn the basic computational techniques required for genuine physical problems. The first chapter collects together (but does not prove) those aspects of Lie group theory which are of importance to differential equations.

This book is devoted to explaining a wide range of applications of con­ tinuous symmetry groups to physically important systems of differential equations. Emphasis is placed on significant applications of group-theoretic methods, organized so that the applied reader can readily learn the basic computational techniques required for genuine physical problems. The first chapter collects together (but does not prove) those aspects of Lie group theory which are of importance to differential equations.