Postnikov M.M. Lectures in Geometry:
Lie Groups and Lie Algebras. Semester 5

Contents

PREFACE

The theory of Lie groups relies on Cartan's theorem on the equivalence of the category of simply connected Lie groups to that of Lie algebras. This book presents the proof of the Cartan theorem and the main results. The branches of the theory of Lie groups which rest on the Cartan theorem remain outside the limits of our exposition. The theory of Lie algebras has been developed to an extent necessary for the Cartan theorem to be proved.

This book like the previous ones of this series* is a nearly faithful record of the Lectures delivered by the author at Moscow University to students (and postgraduates) of the Faculty of Mathematical Mechanics. However, while books I and II were based on Lectures of a compulsory course, this book is a record of an elective course, which makes it essentially different in a number of respects.

Designed for senior and postgraduate students (these Lectures conditionally belong to the fifth semester since students who attended the Lectures were uniformly distributed over all senior courses) the Lectures allowed the * M. M. Postnikov. Lectures In Geometry: Semester i. Analytic Geometry, Mir Publishers, Moscow, 1981), Semester 2. Linear Algebra and Differential Geometry. Mir Publishers, Moscow, 1982). (Referred to as I and II respectively in what follows.) presentation during teaching period of 90 minutes of much more material than had been possible in books I and II intended for first-year students. The volume of the Lectures was increased due to the fact that they became two hours long (120 minutes) while the breaks became shorter and the Lectures continued after the bell had rung. All this almost doubled the actual volume of each Lecture. Of course, with a less intense pace of teaching, under the conditions of, say, a one-year and not a one-semester course, each Lecture virtually extends into a Lecture and a half or even two Lectures. This book, therefore may be better regarded as a record of a one-year elective course (but I managed sometimes - under particularly favourable circumstances - even in one semester), especially since for various reasons it is usually possible to give not more than twelve or thirteen Lectures during a semester, although the curriculum requires eighteen Lectures.

Because of the acute shortage of time, in teaching an elective course, one has more often than in a compulsory course to confine oneself to the mere idea of a proof, leaving the details for the students to prove. It suffices to formulate, with references to the literature, the auxiliary statements from other branches of mathematics and merely to describe the examples illustrating the general theory, leaving their detailed analysis to the students. When, however, a Lecture is committed to paper, it is not necessary to meet these demands, and what is more, all the proofs should be carried out in detail, the examples completely analysed and constructing the "outside" lemmas proved. This sometimes leads to a two or three-fold increase in the volume of a recorded Lecture.

Every Lecturer, presupposing a certain stock of knowledge in his students, is nevertheless compelled to recall at least in short particularly important facts. In written form one has to expand them into a systematic, sometimes rather large, section for the reader's convenience.

This accounts for the surprisingly large volume of some of the Lectures in the book. Yet, with account of the foregoing, each Lecture here is in fact a record of a real Lecture (within which occur self-understood shifts of the initial and terminal pieces of neighbouring Lectures).

All the Lectures virtually break down into five series. The first series (Lectures 1, 2 and 3) introduces, and explains by way of examples, the basic notions: Lie groups, Lie algebras and the Lie algebras of a given Lie group.

The next series (Lectures 4 to 7) is devoted to the "local theory" of Lie groups, Lectures 4 and 6 establish the equivalence of the category of Lie algebras to that of analytic local Lie groups. The necessary algebraic tools are developed in Lecture 5. In Lecture 7, it is proved that analyticity may in fact be assumed without loss of generality. Local subgroups and local factor groups are also considered here.

Extension from the local to the global theory is carried out in Lectures 8, 9 and 10. Lecture 8 presents the theory of coverings (in the sense of Chevalley, i.e. "without paths"). In Lecture 9 a universal covering group is constructed. In Lecture 10 the Cartan theorem is formulated and discussed. No proof of the theorem is constructed, it is only reduced to the Ado theorem on the existence of an exact linear representation for any Lie algebra.

These three series may serve as a miniature course in the theory of Lie groups for beginners.

Lectures 11 and 12 expound subgroups and quotient groups of Lie groups. Lecture 13 is devoted to Clifford algebras and spinor groups. For the first time in educational literature, particular Lie groups G2 and F4 together with the necessary algebraic tools are considered in detail in Lectures 14 to 16.

The last Lectures, 17 to 21, are of a purely algebraic character and are practically independent of all the foregoing material (except for Lecture 20 which stands somewhat by itself). Formally they are devoted to the proof of the Ado theorem, but in fact they comprise a very large fragment of the theory of Lie algebras (Cartan's criteria for solvability and semisimplicity, the Whitehead lemmas, the Weyl and Levi theorems) which is of independent interest as well.

In conclusion I wish topexpress my gratitude to V.L. Popov whose contribution to the improvement of the original manuscript of the book has greatly surpassed the usual duties of an editor. M.M. Postnikov