Introduction to the theory of Lie groups. Translated from the French by Urmie Ray
[For a review of the French original of 1982 see Zbl 0533.22001]. Today, we have many books on the foundations of Lie theory. Among them there are survey texts (in which many statements are given without proofs) and educational texts. Educational texts may be very simple and rather detailed. This book is an educational text, in which practically all statements, which are contained in it, are proved. Therefore it is written for students, who want not only to study the theory of Lie groups and Lie algebras and to pass an examination, but to study the foundation of Lie theory for effectively using it in further scientific work. The book is based on lectures, which were given by the author (1.10.1921--21.7.2016). The author often not only gives references to statements, external to Lie theory, but he describes some subjects and some basic theories (for example, manifolds, coverings, etc.) and also describes some special topics (for example, topological groups, lattices in Lie groups, applications to number theory, a construction of the simply connected covering of \(\mathrm{SL}(2, \mathbf R)\) etc.). Some ``well known'' (to experts) useful details are also explained in this book. In this textbook, only the basic theory of Lie groups and Lie algebras is studied: Lie groups and Lie algebras, Lie subgroups and Lie subalgebras, homomorphisms of Lie algebras and Lie groups. Some information about representations of Lie groups and Lie algebras is given. Also the universal enveloping Lie algebra is described. There is no structural theory in this book and there is no studying of solvable and nilpotent Lie algebras. There is no study of semi-simple Lie algebras and their structure. But the general Lie theory is stated consistently and in details. It is very well suited for beginners. A lot of attention is paid to the exponential map. One of the features of this book, distinguishing it from many other textbooks, is a detailed consideration of linear Lie groups. Only after that the concepts of Lie algebra and of the general Lie group are introduced.
scientific article; zbMATH DE number 6696150