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Lie Groups and Algebraic Groups

Authors: Arkadij L. Onishchik, Ernest B. Vinberg

Publisher: Springer Berlin Heidelberg

Book Series : Springer Series in Soviet Mathematics

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About this book

This book is based on the notes of the authors' seminar on algebraic and Lie groups held at the Department of Mechanics and Mathematics of Moscow University in 1967/68. Our guiding idea was to present in the most economic way the theory of semisimple Lie groups on the basis of the theory of algebraic groups. Our main sources were A. Borel's paper [34], C. ChevalIey's seminar [14], seminar "Sophus Lie" [15] and monographs by C. Chevalley [4], N. Jacobson [9] and J-P. Serre [16, 17]. In preparing this book we have completely rearranged these notes and added two new chapters: "Lie groups" and "Real semisimple Lie groups". Several traditional topics of Lie algebra theory, however, are left entirely disregarded, e.g. universal enveloping algebras, characters of linear representations and (co)homology of Lie algebras. A distinctive feature of this book is that almost all the material is presented as a sequence of problems, as it had been in the first draft of the seminar's notes. We believe that solving these problems may help the reader to feel the seminar's atmosphere and master the theory. Nevertheless, all the non-trivial ideas, and sometimes solutions, are contained in hints given at the end of each section. The proofs of certain theorems, which we consider more difficult, are given directly in the main text. The book also contains exercises, the majority of which are an essential complement to the main contents.

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Table of Contents

Frontmatter

Chapter 1. Lie Groups

Abstract

Here the notions of the differentiable (smooth) manifold, differentiate (smooth) map, direct product of differentiable manifolds, tangent space and the differential of a map (the tangent map) are assumed to be known. Several other notions and theorems on differentiable manifolds will be recalled in the sequel.

Arkadij L. Onishchik, Ernest B. Vinberg

Chapter 2. Algebraic Varieties

Abstract

The objects that occur in this chapter (vector spaces, algebras, algebraic varieties, etc.) are considered over a fixed ground field K. In subsections 1.5–3.3 it is assumed to be algebraically closed¹. Sometimes we require that it be of zero characteristic. The reader, however, would not lose much by restricting himself to the cases K = ℂ or (where the algebraic closedness is not required) K = ℝ. Only these cases are needed for future applications to the Lie group theory and we only consider more general fields in order to elucidate the algebraic nature of the theory discussed.

Arkadij L. Onishchik, Ernest B. Vinberg

Chapter 3. Algebraic Groups

Abstract

The definition of an algebraic group is similar to that of a Lie group, except that differentiable manifolds are replaced by algebraic varieties and differentiable maps by morphisms of algebraic varieties. In this book we will only consider the algebraic groups whose underlying varieties are affine ones. They are called “affine” or “linear” algebraic groups. The difference between arbitrary groups and affine ones is quite essential from the point of view of algebraic geometry and almost indiscernible from the group-theoretical points of view, since the commutator group of any irreducible algebraic group is an affine algebraic group. Besides, the general linear groups and any of their algebraic subgroups are affine algebraic groups. Therefore the affine algebraic groups are the most interesting ones for the Lie group theory. We will simply call them algebraic groups.

Arkadij L. Onishchik, Ernest B. Vinberg

Chapter 4. Complex Semisimple Lie Groups

Abstract

This chapter deals with the most explored section of the theory of Lie groups and Lie algebras. Its main result is the complete classification of connected complex semisimple Lie groups and their irreducible linear representations. This classification is based on the theory of root systems, which because of its numerous applications deserves a special treatment. The theory is axiomatically developed in §2. During the whole chapter (except 1.1°–1.3°) the ground field is ℂ. All the vector spaces and Lie algebras considered are finite-dimensional.

Arkadij L. Onishchik, Ernest B. Vinberg

Chapter 5. Real Semisimple Lie Groups

Abstract

Our study of real semisimple Lie groups and algebras is based on the theory of complex semisimple Lie groups developed in Ch. 4. This is possible because the complexification of a real semisimple Lie algebra is also semisimple (see 1.4.7). However, the correspondence between real and complex semisimple Lie algebras established with the help of the complexification is not one-to-one; any complex semisimple Lie group has at least two non-isomorphic real forms. As it turns out, to describe the real forms of a given complex semisimple Lie algebra g is the same as to classify the involutive automorphisms of g up to conjugacy in Aut g. This classification is easily obtained from the results of 4.4. The global classification of real semisimple Lie groups makes use of the so-called Cartan decomposition of these groups which also plays an important role in various applications of the Lie group theory.

Arkadij L. Onishchik, Ernest B. Vinberg

Chapter 6. Levi Decomposition

Abstract

In this chapter, which owing to its brevity is not divided into sections, we prove Levi’s theorem on the decomposition of an arbitrary Lie algebra into a semidirect sum of a solvable ideal (radical) and a semisimple subalgebra and the theorem on the uniqueness of this decomposition due to A.I. Malcev. Levi’s theorem implies the result which concludes the classical Lie group theory—the existence of a Lie group with an arbitrary given tangent algebra. Next we will consider an analogue of Levi decomposition for algebraic groups.

Arkadij L. Onishchik, Ernest B. Vinberg

Reference Chapter

Abstract

Let G be a simply connected non-commutative simple complex Lie group, g its tangent algebra, W the Weyl group, (α₀,α_l,…,α₁) the extended system of simple roots. Denote by n₀,n_l,…,n₁ the coefficients of the linear relation among α₀, α₁,…,α₁ normed so that n₀ = 1 (see Table 6).

Arkadij L. Onishchik, Ernest B. Vinberg

Backmatter

Title: Lie Groups and Algebraic Groups
Authors: Arkadij L. Onishchik
Ernest B. Vinberg
Copyright Year: 1990
Publisher: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-74334-4
Print ISBN: 978-3-642-74336-8
DOI: https://doi.org/10.1007/978-3-642-74334-4