Compact Lie Groups

By: Mark R. Sepanski

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Hardcover | 1 December 2006

At a Glance

Hardcover
220 Pages

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24.13 x 15.88 x 1.91

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Blending algebra, analysis, and topology, the study of compact Lie groups is one of the most beautiful areas of mathematics and a key stepping stone to the theory of general Lie groups. Assuming no prior knowledge of Lie groups, this book covers the structure and representation theory of compact Lie groups. Included is the construction of the Spin groups, Schur Orthogonality, the Peter-Weyl Theorem, the Plancherel Theorem, the Maximal Torus Theorem, the Commutator Theorem, the Weyl Integration and Character Formulas, the Highest Weight Classification, and the Borel-Weil Theorem. The necessary Lie algebra theory is also developed in the text with a streamlined approach focusing on linear Lie groups.

Key Features are: - Provides an approach that minimizes advanced prerequisites; - Self-contained and systematic exposition requiring no previous exposure to Lie theory; -Advances quickly to the Peter-Weyl Theorem and its corresponding Fourier theory; - Streamlined Lie algebra discussion reduces the differential geometry prerequisite and allows a more rapid transition to the classification and construction of representations - Exercises sprinkled throughout.

This beginning graduate level text, aimed primarily at Lie Groups courses and related topics, assumes familiarity with elementary concepts from group theory, analysis, and manifold theory. Students, research mathematicians, and physicists interested in Lie theory will find this text very useful.

Industry Reviews

From the reviews:

"Groups serve to parameterize the symmetries of mathematical objects and the ways symmetries combine. ... offers students willing to take a few things on faith a vital vista on a subject they may wish to pursue at the graduate level. Summing Up: Recommended. Upper-division undergraduates through professionals." (D. V. Feldman, CHOICE, Vol. v4 (3), November, 2007)

"The representation theory of compact groups is an old and venerable subject. The problems and solutions are now well understood and serve as a guide for the more advanced parts of the representation theory. ... The present one is intended as a textbook within the reach of a good undergraduate student. ... The reading should be pleasant both for students and for teachers preparing a course on the subject." (David A. Renard, Mathematical Reviews, Issue 2008 a)

"This book offers an introduction to the theories of compact Lie groups and of Lie algebras, which is organized in an unusual way. ... For the ambitious reader many exercises are provided." (A. Cap, Monatshefte fuer Mathematik, Vol. 158 (2), October, 2009)

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Non-Fiction Mathematics Groups & Group Theory Algebra Calculus & Mathematical Analysis Geometry Differential & Riemannian Geometry Topology

Preface	p. xi
Compact Lie Groups	p. 1
Basic Notions	p. 1
Manifolds	p. 1
Lie Groups	p. 2
Lie Subgroups and Homomorphisms	p. 2
Compact Classical Lie Groups	p. 4
Exercises	p. 6
Basic Topology	p. 8
Connectedness	p. 8
Simply Connected Cover	p. 10
Exercises	p. 12
The Double Cover of SO(n)	p. 13
Clifford Algebras	p. 14
Spin[subscript n](R) and Pin[subscript n](R)	p. 16
Exercises	p. 18
Integration	p. 19
Volume Forms	p. 19
Invariant Integration	p. 20
Fubini's Theorem	p. 22
Exercises	p. 23
Representations	p. 27
Basic Notions	p. 27
Definitions	p. 27
Examples	p. 28
Exercises	p. 32
Operations on Representations	p. 34
Constructing New Representations	p. 34
Irreducibility and Schur's Lemma	p. 36
Unitarity	p. 37
Canonical Decomposition	p. 39
Exercises	p. 40
Examples of Irreducibility	p. 41
SU(2) and V[subscript n](C[superscript 2])	p. 41
SO(n) and Harmonic Polynomials	p. 42
Spin and Half-Spin Representations	p. 44
Exercises	p. 45
Harmonic Analysis	p. 47
Matrix Coefficients	p. 47
Schur Orthogonality	p. 47
Characters	p. 49
Exercises	p. 52
Infinite-Dimensional Representations	p. 54
Basic Definitions and Schur's Lemma	p. 54
G-Finite Vectors	p. 56
Canonical Decomposition	p. 58
Exercises	p. 59
The Peter-Weyl Theorem	p. 60
The Left and Right Regular Representation	p. 60
Main Result	p. 64
Applications	p. 66
Exercises	p. 69
Fourier Theory	p. 70
Convolution	p. 71
Plancherel Theorem	p. 72
Projection Operators and More General Spaces	p. 77
Exercises	p. 79
Lie Algebras	p. 81
Basic Definitions	p. 81
Lie Algebras of Linear Lie Groups	p. 81
Exponential Map	p. 83
Lie Algebras for the Compact Classical Lie Groups	p. 84
Exercises	p. 86
Further Constructions	p. 88
Lie Algebra Redheaded	p. 88
Lie Subgroups and Subalgebras	p. 91
Covering Redheaded	p. 92
Exercises	p. 93
Abelian Lie Subgroups and Structure	p. 97
Abelian Subgroups and Subalgebras	p. 97
Maximal Tori and Cartan Subalgebras	p. 97
Examples	p. 98
Conjugacy of Cartan Subalgebras	p. 100
Maximal Torus Theorem	p. 102
Exercises	p. 103
Structure	p. 105
Exponential Map Revisited	p. 105
Lie Algebra Structure	p. 108
Commutator Theorem	p. 109
Compact Lie Group Structure	p. 110
Exercises	p. 111
Roots and Associated Structures	p. 113
Root Theory	p. 113
Representations of Lie Algebras	p. 113
Complexification of Lie Algebras	p. 115
Weights	p. 116
Roots	p. 117
Compact Classical Lie Group Examples	p. 118
Exercises	p. 120
The Standard sl(2,C) Triple	p. 123
Cartan Involution	p. 123
Killing Form	p. 124
The Standard sl(2, C) and su(2) Triples	p. 125
Exercises	p. 129
Lattices	p. 130
Definitions	p. 130
Relations	p. 131
Center and Fundamental Group	p. 132
Exercises	p. 134
Weyl Group	p. 136
Group Picture	p. 136
Classical Examples	p. 137
Simple Roots and Weyl Chambers	p. 139
The Weyl Group as a Reflection Group	p. 143
Exercises	p. 146
Highest Weight Theory	p. 151
Highest Weights	p. 151
Exercises	p. 154
Weyl Integration Formula	p. 156
Regular Elements	p. 156
Main Theorem	p. 159
Exercises	p. 162
Weyl Character Formula	p. 163
Machinery	p. 163
Main Theorem	p. 166
Weyl Denominator Formula	p. 168
Weyl Dimension Formula	p. 168
Highest Weight Classification	p. 169
Fundamental Group	p. 170
Exercises	p. 173
Borel-Weil Theorem	p. 176
Induced Representations	p. 176
Complex Structure on G/T	p. 178
Holomorphic Functions	p. 180
Main Theorem	p. 182
Exercises	p. 184
References	p. 187
Index	p. 193
Table of Contents provided by Ingram. All Rights Reserved.

Compact Lie Groups

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Hardcover

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