Contents


 Preface 
 Lecture 1 
Smooth and topological groups. Relaxing the conditions denning Lie groups. Examples of Lie groups. Cayley transformation. Further examples of Lie groups. Connected and arcwise connected spaces and groups. Reduction of any smooth groups to connected groups. Examples of connected Lie groups 
 Lecture 2 
Leftinvariant vector fields. Parallelizability of Lie groups. Integral curves of leftinvariant vector fields and oneparameter subgroups. Lie functor. An example: a group of invertible elements of an associative algebra. Functions with values in an associative algebra. Oneparameter subgroups of the group G (At) 
 Lecture 3 
Matrix Lie groups admitting the Cayley construction. A generalization of the Cayley construction. Groups possessing Inimages. Lie algebras. Examples of Lie algebras. Lie algebras of vector fields. The Lie algebra of a Lie group. An example: the Lie algebra of a group of invertible elements of an associative algebra. Locally isomorphic Lie groups. Local Lie groups. Lie functor on the category of local Lie groups 
 Lecture 4  The exponent of a linear differential operator. A formula for the values of smooth functions in the normal neighbourhood of the identity of a Lie group. A formula for the values of smooth functions on the product of two elements. The convergence of a CampbellHausdorff series. The reconstruction of a local Lie group from its Lie algebra. Operations in the Lie algebra of a Lie group and oneparameter subgroups. Differentials of internal automorphisms. The differential of an exponential mapping. Canonical coordinates. The uniqueness of the structure of a Lie group. Groups without small subgroups and Hilbert's fifth problem 
 Lecture 5 
Free associative algebras. Free Lie algebras. The basic lemma. The universal enveloping algebra. The embedding of a Lie algebra into its universal enveloping algebra. Proof of the fact that the algebra I is free. The PoincareBirkhoffWitt theorem. Tensor products of vector spaces and of algebras. Hopf algebras 
 Lecture 6 
The Friedrichs theorem. The proof of Statement B of 
 Lecture 4. The Dynkin theorem. The linear part of a CampbellHausdorff series. The convergence of a CampbellHausdorff series. Lie group algebras. The equivalence of the categories of local Lie groups and of Lie group algebras. Isomorphism of the categories of Lie group algebras and of Lie algebras. Lie's third theorem 
 Lecture 7 
Local subgroups and subalgebras. Invariant local subgroups and ideals. Local factor groups and quotient algebras. Reducing smooth local groups to analytic ones. Pfafnan systems. Subfiberings of tangent bundles. Integrable subfiberings. Graphs of a Pfaffian system. Involutory subfiberings. The complete univalence of a Lie functor. The involutedness of integrable subfiberings. Completely integrable subfiberings 
 Lecture 8 
Coverings. Sections of coverings. Pointed coverings. Coamalgams. Simply connected spaces. Morphisms of coverings. The relation of quasiorder in the category of pointed coverings. The existence of simply connected coverings. Questions of substantiation. The functorial property of a universal covering 
 Lecture 9 
Smooth coverings. Isomorphism of the categories of smooth and topological coverings. The existence of universal smooth coverings. The coverings of smooth and topological groups. Universal coverings of Lie groups. Lemmas on topological groups. Local isomorphisms and coverings. The description of locally isoraorphic Lie groups 
 Lecture 10 
Local isomorphisms and isomorphisms of localizations. The Cartan theorem. A final diagram of categories and functors. Reduction of the Cartan theorem. The globalizability of embeddable local groups. Reducing the Cartan theorem to the Ado theorem 
 Lecture 11 
Submanifolds of smooth manifolds. Subgroups of Lie groups. Integral manifolds of integrable subfiberings. Maximal integral manifolds. The idea of the proof of Theorem 1. The local structure of Submanifolds. The uniqueness of the structure of a locally rectifiable submanifold with a countable base. Submanifolds of manifolds with a countable base. Connected Lie groups have a countable base. The local rectifiability of maximal integral manifolds. The proof of Theorem 1 
 Lecture 12 
Alternative definitions of a subgroup of a Lie group. Topological subgroups of Lie groups. Closed subgroups of Lie
groups. Algebraic groups. Groups of automorphisms of algebras. Groups of automorphisms of Lie groups. Ideals and invariant subgroups. Quotient manifolds of Lie groups. Quotient groups of Lie groups. The calculation of fundamental groups. The simpleconnectedness of groups SU (n) and Sp (n). The fundamental group of a group 
 Lecture 13 
The Clifford algebra of a quadratic functional, I2graduation of a Clifford algebra. More about tensor multiplication of vector spaces and algebras. Decomposition of Clifford algebras into a skew tensor product. The basis of a Clifford algebra. Conjugation in a Clifford algebra. The centre of a Clifford algebra. A Lie group Spin(n). The fundamental group of a group S0(n). Groups Spin(n) with n << 4. Homomorphism x The group Spin(6). The group Spin(5). Matrix representations of Clifford algebras. Matrix representations of groups Spin(n). Matrix groups in which groups Spin(n) are represented. Reduced representations of groups Spin(n). Additional facts from linear algebra 
 Lecture 14 
Doubling of algebras. Metric algebras. Normed algebras. Automorphisms and differentiations of metric algebras. Differentiations of a doubled algebra. Differentiations and automorphisms of the algebra \ft. The algebra of octaves.
The Lie algebra 02 . Structural constants of the Lie algebra p2C. Representation of the Lie algebra p2C by generators and relations 
 Lecture 15 
Identities in the octave algebra Ca. Subalgebras of the octave algebra Ca. The Lie group G2 . The triplicity principle for the group Spin(8). The analogue of the triplicity principle for the group Spin(9). The Albert algebra A\. The octave projection plane
Contents 
 Lecture 16 
Scalar products in the algebra Al. Automorphisms and differentiations of the algebra Al. Adjoint differentiations of the algebra Al. The Freudenthal theorem. Consequences of the Freudenthal theorem. The Lie group F4.
The Lie algebra f4. The structure of the Lie algebra f4C 
 Lecture 17 
Solvable Lie algebras. The radical of a Lie algebra. Abelian Lie algebras. The centre of a Lie algebra. Nilpotent Lie algebras. The nilradical of a Lie algebra. Linear Lie nilalgebras. The Engel theorem. Criteria of nilpotency. Linear irreducible Lie algebras. Reductive Lie algebras. Linear solvable Lie algebras. The nilpotent radical of a Lie algebra 
 Lecture 18 
Trace functional. Killing's functional. The trace functional of a representation. The Jordan decomposition of a linear operator. The Jordan decomposition of the adjoint operator. The Cartan theorem on linear Lie algebras. Proving Cartan's criterion for the solvability of a Lie algebra. Linear Lie algebras with a nonsingular trace functional. Semisimple Lie algebras. Cartan's criterion for semisimplicity. Casimir operators 
 Lecture 19 
Cohomologies of Lie algebras. The Whitehead theorem. The Fitting decomposition. The generalized Whitehead theorem. The Whitehead lemmas. The Weyl complete reducibility theorem. Extensions of Abelian Lie algebras 
 Lecture 20 
The Levi theorem. Simple Lie algebras and simple Lie groups. Cain and unimodular groups. Schur's lemma. The centre of a simple matrix Lie group. An example of a nonmatrix Lie group. De Rham cohomologies. Cohomologies of the Lie algebras of vector fields. Comparison between the cohomologies of a Lie group and its Lie algebra 
 Lecture 21 
Killing's functional of an ideal. Some properties of differentiations. The radical and nilradical of an ideal. Extension of differentiations to a universal enveloping algebra. Ideals of finite codimension of an enveloping algebra. The radical of an associative algebra. Justification of the inductive step of the construction. The proof of the Ado theorem. Conclusion 
 Supplement to the English Translation. Proof.of the Cartan theorem by V. V. Gorbatsevich) 
 Bibliography Subject Index 
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 firstyear 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 oneyear and not a onesemester 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 oneyear 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 threefold 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 selfunderstood 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