About the book

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 sub-groups 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 adotheorem, but in fact they comprise a very large fragment of the theory of Lie algebras (Cartan’s criteria for solvability and semi simplicity, the whitehead lemmas, the Weyl and Levi theorems) which is of independent interest as well.