Naive Lie Theory

ISBN-10: 0387782141
ISBN-13: 9780387782140
Edition: 2008
Authors: John Stillwell
List price: $54.95 Buy it from $17.36
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Description: Until recently, lie theory has been reserved for practictioners, with no lie theory for mathematical beginners. This book aims to fill that gap and it covers all the basics at a level appropriate for junior/senior level undergraduates.

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Book details

List price: $54.95
Copyright year: 2008
Publisher: Springer
Publication date: 7/24/2008
Binding: Hardcover
Pages: 217
Size: 6.25" wide x 9.25" long x 0.50" tall
Weight: 1.122
Language: English

Until recently, lie theory has been reserved for practictioners, with no lie theory for mathematical beginners. This book aims to fill that gap and it covers all the basics at a level appropriate for junior/senior level undergraduates.

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

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