Naive Lie Theory

ISBN-10: 0387782141

ISBN-13: 9780387782140

Edition: 2008

Authors: John Stillwell

List price: $54.95
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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

Geometry of complex numbers and quaternions
Rotations of the plane
Matrix representation of complex numbers
Consequences of multiplicative absolute value
Quaternion representation of space rotations
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)
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
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
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
Quaternion Lie algebras
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)
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
Open and closed sets in Euclidean space
Closed matrix groups
Continuous functions
Compact sets
Continuous functions and compactness
Paths and path-connectedness
Simple connectedness
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
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