リー代数入門(テキスト)<br>Introduction to Lie Algebras (Springer Undergraduate Mathematics)

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リー代数入門(テキスト)
Introduction to Lie Algebras (Springer Undergraduate Mathematics)

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  • 製本 Paperback:紙装版/ペーパーバック版/ページ数 251 p.
  • 言語 ENG
  • 商品コード 9781846280405
  • DDC分類 512.482

基本説明

The first and only basic introduction to Lie Algebras that's designed specifically for undergraduates; Includes plenty of examples, exercises - with solutions - and problems, making it ideal for independent study.

Full Description


Lie groups and Lie algebras have become essential to many parts of mathematics and theoretical physics, with Lie algebras a central object of interest in their own right.This book provides an elementary introduction to Lie algebras based on a lecture course given to fourth-year undergraduates. The only prerequisite is some linear algebra and an appendix summarizes the main facts that are needed. The treatment is kept as simple as possible with no attempt at full generality. Numerous worked examples and exercises are provided to test understanding, along with more demanding problems, several of which have solutions. Introduction to Lie Algebras covers the core material required for almost all other work in Lie theory and provides a self-study guide suitable for undergraduate students in their final year and graduate students and researchers in mathematics and theoretical physics.

Table of Contents

Preface                                            v
Introduction 1 (10)
Definition of Lie Algebras 1 (1)
Some Examples 2 (1)
Subalgebras and Ideals 3 (1)
Homomorphisms 4 (1)
Algebras 5 (1)
Derivations 6 (1)
Structure Constants 7 (4)
Ideals and Homomorphisms 11 (8)
Constructions with Ideals 11 (1)
Quotient Algebras 12 (2)
Correspondence between Ideals 14 (5)
Low-Dimensional Lie Algebras 19 (8)
Dimensions 1 and 2 20 (1)
Dimension 3 20 (7)
Solvable Lie Algebras and a Rough 27 (10)
Classification
Solvable Lie Algebras 27 (4)
Nilpotent Lie Algebras 31 (1)
A Look Ahead 32 (5)
Subalgebras of gl(V) 37 (8)
Nilpotent Maps 37 (1)
Weights 38 (1)
The Invariance Lemma 39 (3)
An Application of the Invariance Lemma 42 (3)
Engel's Theorem and Lie's Theorem 45 (8)
Engel's Theorem 46 (2)
Proof of Engel's Theorem 48 (1)
Another Point of View 48 (1)
Lie's Theorem 49 (4)
Some Representation Theory 53 (14)
Definitions 53 (1)
Examples of Representations 54 (1)
Modules for Lie Algebras 55 (2)
Submodules and Factor Modules 57 (1)
Irreducible and Indecomposable Modules 58 (2)
Homomorphisms 60 (1)
Schur's Lemma 61 (6)
Representations of sl(2, C) 67 (10)
The Modules Vd 67 (4)
Classifying the Irreducible sl(2, 71 (3)
C)-Modules
Weyl's Theorem 74 (3)
Cartan's Criteria 77 (14)
Jordan Decomposition 77 (1)
Testing for Solvability 78 (2)
The Killing Form 80 (1)
Testing for Semisimplicity 81 (3)
Derivations of Semisimple Lie Algebras 84 (1)
Abstract Jordan Decomposition 85 (6)
The Root Space Decomposition 91 (18)
Preliminary Results 92 (3)
Cartan Subalgebras 95 (2)
Definition of the Root Space Decomposition 97 (1)
Subalgebras Isomorphic to sl(2, C) 97 (2)
Root Strings and Eigenvalues 99 (3)
Cartan Subalgebras as Inner-Product Spaces 102(7)
Root Systems 109(16)
Definition of Root Systems 109(2)
First Steps in the Classification 111(4)
Bases for Root Systems 115(5)
Cartan Matrices and Dynkin Diagrams 120(5)
The Classical Lie Algebras 125(16)
General Strategy 126(3)
sl(l + 1, C) 129(1)
so(2l + 1, C) 130(3)
so(2l, C) 133(1)
sp(2l, C) 134(2)
Killing Forms of the Classical Lie 136(1)
Algebras
Root Systems and Isomorphisms 137(4)
The Classification of Root Systems 141(12)
Classification of Dynkin Diagrams 142(6)
Constructions 148(5)
Simple Lie Algebras 153(10)
Serre's Theorem 154(4)
On the Proof of Serre's Theorem 158(2)
Conclusion 160(3)
Further Directions 163(26)
The Irreducible Representations of a 164(7)
Semisimple Lie Algebra
Universal Enveloping Algebras 171(6)
Groups of Lie Type 177(2)
Kac--Moody Lie Algebras 179(1)
The Restricted Burnside Problem 180(3)
Lie Algebras over Fields of Prime 183(1)
Characteristic
Quivers 184(5)
Appendix A: Linear Algebra 189(20)
Quotient Spaces 189(2)
Linear Maps 191(1)
Matrices and Diagonalisation 192(5)
Interlude: The Diagonal Fallacy 197(1)
Jordan Canonical Form 198(2)
Jordan Decomposition 200(1)
Bilinear Algebra 201(8)
Appendix B: Weyl's Theorem 209(6)
Trace Forms 209(2)
The Casimir Operator 211(4)
Appendix C: Cartan Subalgebras 215(8)
Root Systems of Classical Lie Algebras 215(2)
Orthogonal and Symplectic Lie Algebras 217(3)
Exceptional Lie Algebras 220(1)
Maximal Toral Subalgebras 221(2)
Appendix D: Weyl Groups 223(8)
Proof of Existence 223(1)
Proof of Uniqueness 224(2)
Weyl Groups 226(5)
Appendix E: Answers to Selected Exercises 231(16)
Bibliography 247(2)
Index 249