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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