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Book review

Author(s) Vilenkin, N.Ja.
Klimyk, A.U.
Title Representation of Lie Groups and Special Functions, Vol. 3: Classical and Quantum Groups and Special Functions
Publisher Kluwer Academic Publishers
Year of publication 1992
Reviewed by Valerii Y.TROFIMOV

Contemporary mathematical physics employs very unusual and intricate objects. Quantum groups (Hopf's algebras with special properties), homogeneous spaces of the semisimple Lie groups, theta functions of the Riemannian surfaces, affine Lie algebras (infinite dimensional) is a short list of such notions. The third volume supplies the perfect review of its applications to the theory of special functions. Racah coefficients, Askey-Wilson polynomials, zonal spherical functions, string function and etc. are considered in detail. It might be well to point out that modular forms presented here play important role in recent number theory. For example, the Dirichlet series are the classical objects of this area and we can find their properties in the book under review.

One of the author of the present volume is creator of this direction of the mathematical physics. I think, this three-volume encyclopedia gathers all that is known in this field. The rough contents of the third volume are as follows: 14. Quantum Groups, q-Orthogonal Polynomials and Basic Hypergeometric Functions, 15. Semisimple Lie Groups and Related Homogeneous Spaces, 16. Representations of semisimple Lie Groups and Their Matrix Elements, 17. Group Representations and Special Functions of Matrix Argument, 18. Representations in the Gel'fand-Tsetlin Basis and Special Functions, 19. Modular Forms, Theta Functions and Representations of affine Lie ALgebras.

There are an extensive bibliography (318 titles), list of special symbols (90 items), subject index (162 notions), and bibliography notes to all chapters.

Representation of Lie Groups and Special Functions is corresponded to the grandly conceived plan of the series "Mathematics and Its Applications" (Managing Editor M. Hazewinkel). This encyclopedia is the unique, best reference to the foundations, principles, and applications of the special functions, connected with the quantum groups, semisimple Lie groups and affine Lie algebras. I think, the purchase of this well-written monograph is a good investment for research mathematicians, physicists, engineers, and libraries alike.