李群與李代數III

前言

　　要使我國的數學事業(yè)更好地發(fā)展起來，需要數學家淡泊名利并付出更艱苦地努力。另一方面，我們也要從客觀上為數學家創(chuàng)造更有利的發(fā)展數學事業(yè)的外部環(huán)境，這主要是加強對數學事業(yè)的支持與投資力度，使數學家有較好的工作與生活條件，其中也包括改善與加強數學的出版工作。　　從出版方面來講，除了較好較快地出版我們自己的成果外，引進國外的先進出版物無疑也是十分重要與必不可少的。從數學來說，施普林格（springer）出版社至今仍然是世界上最具權威的出版社。科學出版社影印一批他們出版的好的新書，使我國廣大數學家能以較低的價格購買，特別是在邊遠地區(qū)工作的數學家能普遍見到這些書，無疑是對推動我國數學的科研與教學十分有益的事?！　∵@次科學出版社購買了版權，一次影印了23本施普林格出版社出版的數學書，就是一件好事，也是值得繼續(xù)做下去的事情。大體上分一下，這23本書中，包括基礎數學書5本，應用數學書6本與計算數學書12本，其中有些書也具有交叉性質。這些書都是很新的，2000年以后出版的占絕大部分，共計16本，其余的也是1990年以后出版的。這些書可以使讀者較快地了解數學某方面的前沿，例如基礎數學中的數論、代數與拓撲三本，都是由該領域大數學家編著的“數學百科全書”的分冊。對從事這方面研究的數學家了解該領域的前沿與全貌很有幫助。按照學科的特點，基礎數學類的書以“經典”為主，應用和計算數學類的書以“前沿”為主。這些書的作者多數是國際知名的大數學家，例如《拓撲學》一書的作者諾維科夫是俄羅斯科學院的院士，曾獲“菲爾茲獎”和“沃爾夫數學獎”。這些大數學家的著作無疑將會對我國的科研人員起到非常好的指導作用?！　‘斎?，23本書只能涵蓋數學的一部分，所以，這項工作還應該繼續(xù)做下去。更進一步，有些讀者面較廣的好書還應該翻譯成中文出版，使之有更大的讀者群?！　】傊?，我對科學出版社影印施普林格出版社的部分數學著作這一舉措表示熱烈的支持，并盼望這一工作取得更大的成績。

內容概要

The book contains a comprehensive account of the structure and classification of Lie groups and finite-dimensional Lie algebras (including semisimple，solvable，and of general type). In particular，a modem approach to the description of automorphisms and gradings of semisimple Lie algebras is given. A special chapter is devoted to models of the exceptional Lie algebras. The book contains many tables and will serve as a reference. At the same time many results are accompanied by short proofs.    Onishchik and Vinberg are internationally known specialists in their field; they are also well known for their monograph“Lie Groups and Algebraic Groups”(Springer-Verlag 1990).    The book will be immensely useful to graduate students in differential geometry，algebra and theoretical physics.

書籍目錄

IntroductionChapter 1. General Theorems  1. Lie's and Engel's Theorems    1.1. Lie's Theorem    1.2. Generalizations of Lie's Theorem    1.3. Engel's Theorem and Corollaries to It    1.4. An Analogue of Engel's Theorem in Group Theory  2. The Caftan Criterion    2.1. Invariant Bilinear Forms    2.2. Criteria of Solvability and Semisimplicity    2.3. Factorization into Simple Factors  3. Complete Reducibility of Representations and Triviality of the Cohomology of Semisimple Lie Algebras    3.1. Cohomological Criterion of Complete Reducibility    3.2. The Casimir Operator    3.3. Theorems on the Triviality of Cohomology    3.4. Complete Reducibility of Representations    3.5. Reductive Lie Algebras  4. Levi Decomposition    4.1. Levi's Theorem    4.2. Existence of a Lie Group with a Given Tangent Algebra    4.3. Malcev's Theorem    4.4. Classification of Lie Algebras with a Given Radical  5.Linear Lie Groups    5.1. Basic Notions    5.2. Some Examples    5.3. Ado's Theorem    5.4. Criteria of Linearizability for Lie Groups. Linearizer    5.5. Sufficient Linearizability Conditions    5.6. Structure of Linear Lie Groups  6. Lie Groups and Algebraic Groups    6.1. Complex and Real Algebraic Groups    6.2. Algebraic Subgroups and Subalgebras    6.3. Semisimple and Reductive Algebraic Groups    6.4. Polar Decomposition    6.5. Chevalley Decomposition  7. Complexification and Real Forms    7.1. Complexification and Real Forms of Lie Algebras    7.2. Complexification and Real Forms of Lie Groups    7.3. Universal Complexification of a Lie Group  8. Splittings of Lie Groups and Lie Algebras    8.1. Malcev Splittable Lie Groups and Lie Algebras    8.2. Definition of Splittings of Lie Groups and Lie Algebras    8.3. Theorem on the Existence and Uniqueness of Splittings  9. Caftan Subalgebras and Subgroups. Weights and Roots    9.1. Representations of Nilpotent Lie Algebras    9.2. Weights and Roots with Respect to a Nilpotent Subalgebra    9.3. Caftan Subalgebras    9.4. Caftan Subalgebras and Root Decompositions of Semisimple Lie Algebras    9.5. Caftan SubgroupsChapter 2. Solvable Lie Groups and Lie Algebras  1. Examples  2. Triangular Lie Groups and Lie Algebras  3. Topology of Solvable Lie Groups and Their Subgroups    3.1. Canonical Coordinates    3.2. Topology of Solvable Lie Groups    3.3. Aspherical Lie Groups    3.4. Topology of Subgroups of Solvable Lie Groups  4. Nilpotent Lie Groups and Lie Algebras    4.1. Definitions and Examples    4.2. Malcev Coordinates    4.3. Cohomology and Outer Automorphisms  5. Nilpotent Radicals in Lie Algebras and Lie Groups    5.1. Nilradical    5.2. Nilpotent Radical    5.3. Unipotent Radical  6. Some Classes of Solvable Lie Groups and Lie Algebras    6.1. Characteristically Nilpotent Lie Algebras    6.2. Filiform Lie Algebras    6.3. Nilpotent Lie Algebras of Class 2    6.4. Exponential Lie Groups and Lie Algebras    6.5. Lie Algebras and Lie Groups of Type (I)  7. Linearizability Criterion for Solvable Lie GroupsChapter 3. Complex Semisimple Lie Groups and Lie Algebras  1. Root Systems    1.1. Abstract Root Systems    1.2. Root Systems of Reductive Groups    1.3. Root Decompositions and Root Systems for Classical Complex Lie Algebras    1.4. Weyl Chambers and Simple Roots    1.5. Borel Subgroups and Subalgebras    1.6. The Weyl Group    1.7. The Dynkin Diagram and the Cartan Matrix    1.8. Classification of Admissible Systems of Vectors and Root Systems    1.9. Root and Weight Lattices    1.10. Chevalley Basis  2. Classification of Complex Semisimple Lie Groups and Their Linear Representations    2.1. Uniqueness Theorems for Lie Algebras    2.2. Uniqueness Theorem for Linear Representations    2.3. Existence Theorems    2.4. Global Structure of Connected Semisimple Lie Groups    2.5. Classification of Connected Semisimple Lie Groups    2.6. Linear Representations of Connected Reductive Algebraic Groups    2.7. Dual Representations and Bilinear Invariants    2.8. The Kernel and the Image of a Locally Faithful Linear Representation    2.9. The Casimir Operator and Dynkin Index    2.10. Spinor Group and Spinor Representation  3. Automorphisms and Gradings    3.1. Description of the Group of Automorphisms    3.2. Quasitori of Automorphisms and Gradings    3.3. Homogeneous Semisimple and Nilpotent Elements    3.4. Fixed Points of Automorphisms    3.5. One-dimensional Tori of Automorphisms and Z-gradings    3.6. Canonical Form of an Inner Semisimple Automorphism    3.7. Inner Automorphisms of Finite Order and Zm-gradings of Inner Type    3.8. Quasitorus Associated with a Component of the Group of Automorphisms    3.9. Generalized Root Decomposition    3.10. Canonical Form of an Outer Semisimple Automorphism    3.11. Outer Automorphisms of Finite Order and Zm-gradings of Outer Type    3.12. Jordan Gradings of Classical Lie Algebras    3.13. Jordan Gradings of Exceptional Lie AlgebrasChapter 4. Real Semisimple Lie Groups and Lie Algebras  1. Classification of Real Semisimple Lie Algebras    1.1. Real Forms of Classical Lie Groups and Lie Algebras    1.2. Compact Real Form    1.3. Real Forms and Involutory Automorphisms    1.4. Involutory Automorphisms of Complex Simple Algebras    1.5. Classification of Real Simple Lie Algebras  2. Compact Lie Groups and Complex Reductive Groups    2.1. Some Properties of Linear Representations of Compact Lie Groups    2.2. Selfoadjointness of Reductive Algebraic Groups    2.3. Algebralcity of a Compact Lie Group    2.4. Some Properties of Extensions of Compact Lie Groups    2.5. Correspondence Between Real Compact and Complex Reductive Lie Groups    2.6. Maximal Tori in Compact Lie Groups  3. Cartan Decomposition    3.1. Cartan Decomposition of a Semisimple Lie Algebra    3.2. Caftan Decomposition of a Semisimple Lie Group    3.3. Conjugacy of Maximal Compact Subgroups of Semisimple Lie Groups    3.4. Topological Structure of Lie Groups    3.5. Classification of Connected Semisimple Lie Groups    3.6. Linearizer of a Semisimple Lie Group  4. Real Root Decomposition    4.1. Maximal R-Diagonalizable Subalgebras    4.2. Real Root Systems    4.3. Satake Diagrams    4.4. Split Real Semisimple Lie Algebras    4.5. Iwasawa Decomposition    4.6. Maximal Connected Triangular Subgroups    4.7. Cartan Subalgebras of a Real Semisimple Lie Algebra  5. Exponential Mapping for Semisimple Lie Groups    5.1. Image of the Exponential Mapping    5.2. Index of an Element of a Lie Group    5.3. Indices of Simple Lie GroupsChapter 5. Models of Exceptional Lie Algebras  1. Models Associated with the Cayley Algebra    1.1, Cayley Algebra    1.2. The Algebra G2    1.3. Exceptional Jordan Algebra    1.4. The Algebra F4    1.5. The Algebra E6    1.6. The Algebra E7    1.7. Unified Construction of Exceptional Lie Algebras  2. Models Associated with GradingsChapter 6. Subgroups and Subalgebras of Semisimple Lie Groups and Lie Algebras  1. Regular Subalgebras and Subgroups    1.1. Regular Subalgebras of Complex Semisimple Lie Algebras    1.2. Description of Semisimple and Reductive Regular Subalgebras    1.3. Parabolic Subalgebras and Subgroups    1.4. Examples of Parabolic Subgroups and Flag Manifolds    1.5. Parabolic Subalgebras of Real Semisimple Lie Algebras    1.6. Nonsemisimple Maximal Subalgebras  2. Three-dimensional Simple Subalgebras and Nilpotent Elements    2.1. sι2-triples    2.2. Three-dimensional Simple Subalgebras of Classical Simple Lie Algebras    2.3. Principal and Semiprincipal Three-dimensional Simple Subalgebras    2.4. Minimal Ambient Regular Subalgebras    2.5. Minimal Ambient Complete Regular Subalgebras  3. Semisimple Subalgebras and Subgroups    3.1. Semisimple Subgroups of Complex Classical Groups    3.2. Maximal Connected Subgroups of Complex Classical Groups    3.3. Semisimple Subalgebras of Exceptional Complex Lie Algebras    3.4. Semisimple Subalgebras of Real Semisimple Lie AlgebrasChapter 7. On the Classification of Arbitrary Lie Groups and Lie Algebras of a Given Dimension  1. Classification of Lie Groups and Lie Algebras of Small Dimension    1.1. Lie Algebras of Small1 Dimension    1.2. Connected Lie Groups of Dimension < 3  2. The Space of Lie Algebras. Deformations and Contractions    2.1. The Space of Lie Algebras    2.2. Orbits of the Action of the Group Gιn(k) on ι(k)    2.3. Deformations of Lie Algebras    2.4. Rigid Lie Algebras    2.5. Contractions of Lie Algebras    2.6. Spaces ιn(k) for Small nTablesReferencesAuthor IndexSubject Index

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