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| Geometry of complex numbers and quaternions | |
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| Rotations of the plane | |
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| Matrix representation of complex numbers | |
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| Quaternions | |
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| Consequences of multiplicative absolute value | |
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| Quaternion representation of space rotations | |
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| Discussion | |
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| Groups | |
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| Crash course on groups | |
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| Crash course on homomorphisms | |
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| The groups SU(2) and SO(3) | |
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| Isometries of R[superscript n] and reflections | |
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| Rotations of R[superscript 4] and pairs of quaternions | |
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| Direct products of groups | |
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| The map from SU(2)xSU(2) to SO(4) | |
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| Discussion | |
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| Generalized rotation groups | |
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| Rotations as orthogonal transformations | |
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| The orthogonal and special orthogonal groups | |
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| The unitary groups | |
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| The symplectic groups | |
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| Maximal tori and centers | |
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| Maximal tori in SO(n), U(n), SU(n), Sp(n) | |
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| Centers of SO(n), U(n), SU(n), Sp(n) | |
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| Connectedness and discreteness | |
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| Discussion | |
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| The exponential map | |
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| The exponential map onto SO(2) | |
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| The exponential map onto SU(2) | |
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| The tangent space of SU(2) | |
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| The Lie algebra su(2) of SU(2) | |
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| The exponential of a square matrix | |
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| The affine group of the line | |
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| Discussion | |
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| The tangent space | |
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| Tangent vectors of O(n), U(n), Sp(n) | |
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| The tangent space of SO(n) | |
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| The tangent space of U(n), SU(n), Sp(n) | |
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| Algebraic properties of the tangent space | |
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| Dimension of Lie algebras | |
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| Complexification | |
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| Quaternion Lie algebras | |
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| Discussion | |
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| Structure of Lie algebras | |
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| Normal subgroups and ideals | |
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| Ideals and homomorphisms | |
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| Classical non-simple Lie algebras | |
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| Simplicity of sl(n, C) and su(n) | |
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| Simplicity of so(n) for n > 4 | |
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| Simplicity of sp(n) | |
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| Discussion | |
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| The matrix logarithm | |
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| Logarithm and exponential | |
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| The exp function on the tangent space | |
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| Limit properties of log and exp | |
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| The log function into the tangent space | |
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| SO(n), SU(n), and Sp(n) revisited | |
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| The Campbell-Baker-Hausdorff theorem | |
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| Eichler's proof of Campbell-Baker-Hausdorff | |
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| Discussion | |
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| Topology | |
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| Open and closed sets in Euclidean space | |
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| Closed matrix groups | |
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| Continuous functions | |
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| Compact sets | |
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| Continuous functions and compactness | |
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| Paths and path-connectedness | |
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| Simple connectedness | |
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| Discussion | |
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| Simply connected Lie groups | |
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| Three groups with tangent space R | |
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| Three groups with the cross-product Lie algebra | |
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| Lie homomorphisms | |
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| Uniform continuity of paths and deformations | |
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| Deforming a path in a sequence of small steps | |
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| Lifting a Lie algebra homomorphism | |
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| Discussion | |
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| Bibliography | |
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| Index | |