| 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 |
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