樸素李理論

內容概要

The existence of these little—known elementary proofs convinced me that a naive approach to Lie theory is possible and desirable． The aim of this book is to carry it outdeveloping the central concepts and results of Lie theory by the simplest possible methods， mainly from single—variable calculus and linear algebra． Familiarity with elementary group theory is also desirable， but I provide a crash course on the basics of group theory in Sections 2．1 and 2．2．

書籍目錄

1 Geometry of complex numbers and quaternions
　1.1 Rotations of the plane
　1.2 Matrix representation of complex numbers
　1.3 Quaternions
　1.4 Consequences of multiplicative absolute value
　1.5 Quaternion representation of space rotations
　1.6 Discussion
2 Groups
　2.1 Crash course on groups
　2.2 Crash course on homomorphisms
　2.3 The groups SU(2) and SO(3)
　2.4 Isometries of * and reflections
　2.5 Rotations of R4 and pairs of quaternions
　2.6 Direct products of groups
　2.7 The map from SU(2)xSU(2) to SO(4)
　2.8 Discussion
3 Generalized rotation groups
　3.1 Rotations as orthogonal transformations
　3.2 The orthogonal and special orthogonal groups
　3.3 The unitary groups
　3.4 The symplectic groups
　3.5 Maximal toil and centers
　3.6 Maximal toil in SO(n), U(n), SU(n), Sp(n)
　3.7 Centers of SO(n), U(n), SU(n), Sp(n)
　3.8 Connectedness and discreteness
　3.9 Discussion
　……

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