Matrix groups : an introduction to lie group theory矩阵小组
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作者： Andrew Baker著
出 版 社：
出版时间： 2003-10-1字数：版次：页数： 330印刷时间： 2003/10/01开本： 16开印次：纸张： 胶版纸I S B N ： 9781852334703包装： 平装内容简介
Aimed at advanced undergraduate and beginning graduate students, this book provides a first taste of the theory of Lie groups as an appetiser for a more substantial further course. Lie theoretic ideas lie at the heart of much of standard undergraduate linear algebra and exposure to them can inform or motivate the study of the latter.
The main focus is on matrix groups, i.e., closed subgroups of real and complex general linear groups. The first part studies examples and describes the classical families of simply connected compact groups. The second part introduces the idea of a lie group and studies the associated notion of a homogeneous space using orbits of smooth actions.
Throughout, the emphasis is on providing an approach that is accessible to readers equipped with a standard undergraduate toolkit of algebra and analysis. Although the formal prerequisites are kept as low level as possible, the subject matter is sophisticated and contains many of the key themes of the fully developed theory, preparing students for a more standard and abstract course in Lie theory and differential geometry.
目录
Part I. Basic Ideas and Examples
1. Real and Complex Matrix Groups
1.1 Groups of Matrices
1.2 Groups of Matrices as Metric Spaces
1.3 Compactness
1.4 Matrix Groups
1.5 Some Important Examples
1.6 Complex Matrices as Real Matrices
1.7 Continuous Homomorphisms of Matrix Groups
1.8 Matrix Groups for Normed Vector Spaces
1.9 Continuous Group Actions
2. Exponentials, Differential Equations and One-parameter Sub- groups
2.1 The Matrix Exponential and Logarithm
2.2 Calculating Exponentials and Jordan Form
2.3 Differential Equations in Matrices
2.4 One-parameter Subgroups in Matrix Groups
2.5 One-parameter Subgroups and Differential Equations
3. Tangent Spaces and Lie Algebras
3.1 Lie Algebras
3.2 Curves, Tangent Spaces and Lie Algebras
3.3 The Lie Algebras of Some Matrix Groups
3.4. Some Observations on the Exponential Function of a Matrix Group
3.5 SO(3) and SU(2)
3.6 The Complexification of a Real Lie Algebra
4. Algebras, Quaternions and Quaternionic Symplectic Groups
4.1 Algebras
4.2 Real and Complex Normed Algebras
4.3 Linear Algebra over a Division Algebra
4.4 The Quaternions
4.5 Quaternionic Matrix Groups
4.6 Automorphism Groups of Algebras
5. Clifford Algebras and Spluor Groups
5.1 Real Clifford Algebras
5.2 Clifford Groups
5.3 Pinor and Spinet Groups
5.4 The Centres of Spinor Groups
5.5 Finite Subgroups of Spinor Groups
6. Loreutz Groups
6.1 Lorentz Groups
6.2 A Principal Axis Theorem for Lorentz Groups
6.3 SLy(C) and the Lorentz Group Lot(3,1)
Part II. Matrix Groups as Lie Groups
7. Lie Groups
7.1 Smooth Manifolds
7.2 Tangent Spaces and Derivatives
7.3 Lie Groups
7.4 Some Examples of Lie Groups
7.5 Some Useful Formulae in Matrix Groups
7.6 Matrix Groups are Lie Groups
7.7 Not All Lie Groups are Matrix t~roup~
8. Homogeneous Spaces
8.1 Homogeneous Spaces as Manifolds
8.2 Homogeneous Spaces as Orbits
8.3 Projective Spaces
8.4 Grassmannians
……
Part III.Compact Connected Lie Groups and their Classification
Hints and Solutions to Selected Exercises
Bibliography
Index
