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Basic Ideas and Examples
Real and Complex Matrix Groupsp. 3
Groups of Matricesp. 3
Groups of Matrices as Metric Spacesp. 5
Compactnessp. 12
Matrix Groupsp. 15
Some Important Examplesp. 17
Complex Matrices as Real Matricesp. 29
Continuous Homomorphisms of Matrix Groupsp. 31
Matrix Groups for Normed Vector Spacesp. 33
Continuous Group Actionsp. 37
Exponentials, Differential Equations and One-parameter Subgroupsp. 45
The Matrix Exponential and Logarithmp. 45
Calculating Exponentials and Jordan Formp. 51
Differential Equations in Matricesp. 55
One-parameter Subgroups in Matrix Groupsp. 56
One-parameter Subgroups and Differential Equationsp. 59
Tangent Spaces and Lie Algebrasp. 67
Lie Algebrasp. 67
Curves, Tangent Spaces and Lie Algebrasp. 71
The Lie Algebras of Some Matrix Groupsp. 76
Some Observations on the Exponential Function of a Matrix Groupp. 84
SO(3) and SU(2)p. 86
The Complexification of a Real Lie Algebrap. 92
Algebras, Quaternions and Quaternionic Symplectic Groupsp. 99
Algebrasp. 99
Real and Complex Normed Algebrasp. 111
Linear Algebra over a Division Algebrap. 113
The Quaternionsp. 116
Quaternionic Matrix Groupsp. 120
Automorphism Groups of Algebrasp. 122
Clifford Algebras and Spinor Groupsp. 129
Real Clifford Algebrasp. 130
Clifford Groupsp. 139
Pinor and Spinor Groupsp. 143
The Centres of Spinor Groupsp. 151
Finite Subgroups of Spinor Groupsp. 152
Lorentz Groupsp. 157
Lorentz Groupsp. 157
A Principal Axis Theorem for Lorentz Groupsp. 165
SL[subscript 2](C) and the Lorentz Group Lor(3, 1)p. 171
Matrix Groups as Lie Groups
Lie Groupsp. 181
Smooth Manifoldsp. 181
Tangent Spaces and Derivativesp. 183
Lie Groupsp. 187
Some Examples of Lie Groupsp. 189
Some Useful Formulae in Matrix Groupsp. 193
Matrix Groups are Lie Groupsp. 199
Not All Lie Groups are Matrix Groupsp. 203
Homogeneous Spacesp. 211
Homogeneous Spaces as Manifoldsp. 211
Homogeneous Spaces as Orbitsp. 215
Projective Spacesp. 217
Grassmanniansp. 222
The Gram-Schmidt Processp. 224
Reduced Echelon Formp. 226
Real Inner Productsp. 227
Symplectic Formsp. 229
Connectivity of Matrix Groupsp. 235
Connectivity of Manifoldsp. 235
Examples of Path Connected Matrix Groupsp. 238
The Path Components of a Lie Groupp. 241
Another Connectivity Resultp. 244
Compact Connected Lie Groups and their Classification
Maximal Tori in Compact Connected Lie Groupsp. 251
Torip. 251
Maximal Tori in Compact Lie Groupsp. 255
The Normaliser and Weyl Group of a Maximal Torusp. 259
The Centre of a Compact Connected Lie Groupp. 262
Semi-simple Factorisationp. 267
An Invariant Inner Productp. 267
The Centre and its Lie Algebrap. 270
Lie Ideals and the Adjoint Actionp. 272
Semi-simple Decompositionsp. 276
The Structure of the Adjoint Representationp. 278
Roots Systems, Weyl Groups and Dynkin Diagramsp. 289
Inner Products and Dualityp. 289
Roots systems and their Weyl groupsp. 291
Some Examples of Root Systemsp. 293
The Dynkin Diagram of a Root Systemp. 297
Irreducible Dynkin Diagramsp. 298
From Root Systems to Lie Algebrasp. 299
Hints and Solutions to Selected Exercisesp. 303
Bibliographyp. 323
Indexp. 325
Table of Contents provided by Rittenhouse. All Rights Reserved.

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.