Book review

The book appears as Volume 20 in the series "Encyclopaedia of Mathematical Sciences" (editor-in-chief: R. V. Gamkrelidze) which is in print by Springer-Verlag. The purpose of the series is to give up-to-date reviews of various domains of mathematics. The Springer publishers are fairly experienced in that job: the previous generation of scholars benefited greatly from "yellow books" of "Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen" published between the two World Wars and including a number of classical monographs by the great mathematicians of that epoch. Most volumes published in the present series are updated translations from Russian publications of the USSR Institute of Scientific Information (Moscow), which appeared in a limited number of copies in the series "Modern Problems in Mathematics", as a part of a huge project "Progress in Science and Technology". In spite of a modest presentation and quite a restricted circulation, that Russian mathematical series was famous for decades because of a high level and the reputation of its contributors.

The present volume is a respectable member of its family. It consists of two parts: "Foundations of Lie Theory" by A. L. Onishchik and E. B. Vinberg, and "Lie Transformation Groups" by V. V. Gorbatsevich and A. L. Onishchik. Onishchik and Vinberg have been well known for a long time as leading figures in the Moscow school of group theory. Both have been submitting their works for almost four decades. Gorbatsevich has a research history of about 20 years. A number of the results reported in the book have been obtained by the authors originally.

The first part starts from basic notions and definitions, yet it is neither a textbook, nor a treatise of the Lie theory. There is a lot of such presentations, some of them written by the classics, including dozens of excellent books and review articles on applications of group theory in modern theoretical physics. This short (about 90 pp.) text is rather an elegant compilation of fundamental concepts and results. Brief proofs are given for the most important theorems. Some statements of marginal relevance are presented without proofs, and proper references are given. Connectedness of Lie groups, their general homotopical properties, the relation between Lie groups and Lie algebras, and the universal enveloping algebras are considered in particular in the first three chapters. Chapter 4 is special; it is a survey of some generalizations of Lie groups. In particular, the authors introduced Lie groups on valued fields and the concept of formal groups, described elements of the theory of infinite-dimensional Lie groups and analytic loops (a non-associative generalization of Lie groups). More recent and extensive developments like graded Lie algebras (Lie 'super-algebras'), Kac-Moody algebras and their associated groups, as well as the the so-called q-deformations ('quantum groups') were beyond the scope of the work, as they probably deserve separate and more comprehensive expositions.

The second part presents basic aspects of the theory of continuous transformations of differentiable manifolds. Transitive actions of Lie groups and the actions of compact Lie groups are considered in particular. Homogeneous spaces of nilpotent and solvable groups ('nilmanifolds' and 'solvmanifolds', respectively) are the subject of a special chapter. Another chapter is devoted to compact homogeneous spaces and their topological properties. A number of classification theorems is given. Properties of orbits and stabilizers, homogeneous fibre bundles, Frobenius duality, the problem of slices are considered as well. A complete description of actions of Lie groups in some low-dimensional manifolds is given in the last chapter.

The Russian edition was published in 1988, and some recent references have been added. The translation has been prepared by A. Kozlowski very carefully, but sometimes the Russian original was followed too closely, which makes the English text heavier than it could be, even ambiguous at some points.

This concise and clear presentation of fundamentals of Lie theory is absolutely indispensable for everybody who would have a chance to work with it, especially in applying the methods of Lie groups in various fields of mathematics and theoretical sciences.