Lie Groups and Algebraic Groups

Extraordinary book based on Moscow University hectographed notes from 1968. The book is substantially extended, with material developed through posing of problems and a wealth of examples. Several tables have never been published before, e.g. decomposition of representations into irreducible components. Especially interesting for physicists.

Klappentext

This is a quite extraordinary book on Lie groups and algebraic groups. Created from hectographed notes in Russian from Moscow University, which for many Soviet mathematicians have been something akin to a "bible", the book has been substantially extended and organized to develop the material through the posing of problems and to illustrate it through a wealth of examples. Several tables have never before been published, such as decomposition of representations into irreducible components. This will be especially helpful for physicists. The authors have managed to present some vast topics: the correspondence between Lie groups and Lie algebras, elements of algebraic geometry and of algebraic group theory over fields of real and complex numbers, the main facts of the theory of semisimple Lie groups (real and complex, their local and global classification included) and their representations. The literature on Lie group theory has no competitors to this book in broadness of scope. The book is self-contained indeed: only the very basics of algebra, calculus and smooth manifold theory are really needed. This distinguishes it favorably from other books in the area. It is thus not only an indispensable reference work for researchers but also a good introduction for students.

Inhalt

1. Lie Groups.- § 1. Background.- §2. Tangent Algebra.- §3. Connectedness and Simple Connectedness.- § 4. The Derived Algebra and the Radical.- 2. Algebraic Varieties.- §1. Affine Algebraic Varieties.- § 2. Projective and Quasiprojective Varieties.- § 3. Dimension and Analytic Properties of Algebraic Varieties.- 3. Algebraic Groups.- § 1. Background.- §2. Commutative and Solvable Algebraic Groups.- § 3. The Tangent Algebra.- §4. Compact Linear Groups.- 4. Complex Semisimple Lie Groups.- §1. Preliminaries.- §2. Root Systems.- §3. Existence and Uniqueness Theorems.- §4. Automorphisms.- 5. Real Semisimple Lie Groups.- § 1. Real Forms of Complex Semisimple Lie Groups and Algebras.- § 2. Compact Lie Groups and Reductive Algebraic Groups.- § 3. Cartan Decomposition.- § 4. Real Root Decomposition.- 6. Levi Decomposition.- 1°. Levi's Theorem.- 2°. Existence of a Lie Group with the Given Tangent Algebra.- 3°. Malcev's Theorem.- 4°. Algebraic Levi Decomposition.- Exercises.- Hints to Problems.- Reference Chapter.- § 1. Useful Formulae.- §2. Tables.