1Geometryofcomplexnumbersandquaternions
1.1Rotationsoftheplane
1.2Matrixrepresentationofcomplexnumbers
1.3Quaternions
1.4Consequencesofmultiplicativeabsolutevalue
1.5Quaternionrepresentationofspacerotations
1.6Discussion
2Groups
2.1Crashcourseongroups
2.2Crashcourseonhomomorphisms
2.3ThegroupsSU(2)andSO(3)
2.4IsometricsofR''andreflections
2.5RotationsofR4andpairsofquaternions
2.6Directproductsofgroups
2.7ThemapfromSU(2)SU(2)toSO(4)
2.8Discussion
3Generalizedrotationgroups
3.1Rotationsasorthogonaltransformations
3.2Theorthogonaiandspecialorthogonalgroups
3.3Theunitarygroups
3.4Thesymplecticgroups
3.5Maximaltoriandcenters
3.6MaximaltoriinSO(n),U(n),SU(n),Sp(n)
3.7CentersofSO(n),U(n),SU(n),Sp(n)
3.8Connectcdnessanddiscreteness
3.9Discussion
4Theexponentialmap
4.1TheexponentialmapontoSO(2)
4.2TheexponentialmapontoSU(2)
4.3ThetangentspaceofSU(2)
4.4TheLiealgebrasu(2)ofSU(2)
4.5Theexponentialofasquarematrix
4.6Theaffinegroupoftheline
4.7Discussion
5Thetangentspace
5.1TangentvectorsofO(n),U(n),Sp(n)
5.2ThetangentspaceofSO(n)
5.3ThetangentspaceofU(n),SU(n),Sp(n)
5.4Algebraicpropertiesofthetangentspace
5.5DimensionofLiealgebras
5.6Complexification
5.7QuaternionLiealgebras
5.8Discussion
6StructureofLiealgebras
6.1Normalsubgroupsandideals
6.2Idealsandhomomorphisms
6.3Classicalnon-simpleLiealgebras
6.4Simplicityof(n,C)andsu(n)
6.5Simplicityofo(n)forn>4
6.6Simplicityofp(n)
6.7Discussion
7Thematrixlogarithm
7.1Logarithmandexponential
7.2Theexpfunctiononthetangentspace
7.3Limitpropertiesoflogandexp
7.4Thelogfunctionintothetangentspace
7.5SO(n),SU(n),andSp(n)revisited
7.6TheCampbell-Baker-Hausdorfftheorem
7.7Eichler'sproofofCampbell-Baker-Hausdorff
7,8Discussion
8Topology
8.1OpenandclosedsetsinEuclideanspace
8.2Closedmatrixgroups
8.3Continuousfunctions
8.4Compactsets
8.5Continuousfunctionsandcompactness
8.6Pathsandpath-connectedness
8.7Simpleconnectedness
8.8Discussion
9SimplyconnectedLiegroups
9.1ThreegroupswithtangentspaceR
9.2Threegroupswiththecross-productLiealgebra
9.3Liehomomorphisms
9.4Uniformcontinuityofpathsanddeformations
9.5Deformingapathinasequenceofsmallsteps
9.6LiftingaLiealgebrahomomorphism
9.7Discussion
Bibliography
Index