| Geometry of complex numbers and quaternions | p. 1 |
| Rotations of the plane | p. 2 |
| Matrix representation of complex numbers | p. 5 |
| Quaternions | p. 7 |
| Consequences of multiplicative absolute value | p. 11 |
| Quaternion representation of space rotations | p. 14 |
| Discussion | p. 18 |
| Groups | p. 23 |
| Crash course on groups | p. 24 |
| Crash course on homomorphisms | p. 27 |
| The groups SU(2) and SO(3) | p. 32 |
| Isometries of R[superscript n] and reflections | p. 36 |
| Rotations of R[superscript 4] and pairs of quaternions | p. 38 |
| Direct products of groups | p. 40 |
| The map from SU(2)xSU(2) to SO(4) | p. 42 |
| Discussion | p. 45 |
| Generalized rotation groups | p. 48 |
| Rotations as orthogonal transformations | p. 49 |
| The orthogonal and special orthogonal groups | p. 51 |
| The unitary groups | p. 54 |
| The symplectic groups | p. 57 |
| Maximal tori and centers | p. 60 |
| Maximal tori in SO(n), U(n), SU(n), Sp(n) | p. 62 |
| Centers of SO(n), U(n), SU(n), Sp(n) | p. 67 |
| Connectedness and discreteness | p. 69 |
| Discussion | p. 71 |
| The exponential map | p. 74 |
| The exponential map onto SO(2) | p. 75 |
| The exponential map onto SU(2) | p. 77 |
| The tangent space of SU(2) | p. 79 |
| The Lie algebra su(2) of SU(2) | p. 82 |
| The exponential of a square matrix | p. 84 |
| The affine group of the line | p. 87 |
| Discussion | p. 91 |
| The tangent space | p. 93 |
| Tangent vectors of O(n), U(n), Sp(n) | p. 94 |
| The tangent space of SO(n) | p. 96 |
| The tangent space of U(n), SU(n), Sp(n) | p. 99 |
| Algebraic properties of the tangent space | p. 103 |
| Dimension of Lie algebras | p. 106 |
| Complexification | p. 107 |
| Quaternion Lie algebras | p. 111 |
| Discussion | p. 113 |
| Structure of Lie algebras | p. 116 |
| Normal subgroups and ideals | p. 117 |
| Ideals and homomorphisms | p. 120 |
| Classical non-simple Lie algebras | p. 122 |
| Simplicity of sl(n, C) and su(n) | p. 124 |
| Simplicity of so(n) for n > 4 | p. 127 |
| Simplicity of sp(n) | p. 133 |
| Discussion | p. 137 |
| The matrix logarithm | p. 139 |
| Logarithm and exponential | p. 140 |
| The exp function on the tangent space | p. 142 |
| Limit properties of log and exp | p. 145 |
| The log function into the tangent space | p. 147 |
| SO(n), SU(n), and Sp(n) revisited | p. 150 |
| The Campbell-Baker-Hausdorff theorem | p. 152 |
| Eichler's proof of Campbell-Baker-Hausdorff | p. 154 |
| Discussion | p. 158 |
| Topology | p. 160 |
| Open and closed sets in Euclidean space | p. 161 |
| Closed matrix groups | p. 164 |
| Continuous functions | p. 166 |
| Compact sets | p. 169 |
| Continuous functions and compactness | p. 171 |
| Paths and path-connectedness | p. 173 |
| Simple connectedness | p. 177 |
| Discussion | p. 182 |
| Simply connected Lie groups | p. 186 |
| Three groups with tangent space R | p. 187 |
| Three groups with the cross-product Lie algebra | p. 188 |
| Lie homomorphisms | p. 191 |
| Uniform continuity of paths and deformations | p. 194 |
| Deforming a path in a sequence of small steps | p. 195 |
| Lifting a Lie algebra homomorphism | p. 197 |
| Discussion | p. 201 |
| Bibliography | p. 204 |
| Index | p. 207 |
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