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  • 书号:9787030313911
    作者:Mark R.Sepanski
  • 外文书名:Compact Lie Groups
  • 装帧:
    开本:B5
  • 页数:220
    字数:253
    语种:
  • 出版社:科学出版社
    出版时间:2011/6/28
  • 所属分类:O15 代数、数论、组合理论
  • 定价: ¥65.00元
    售价: ¥51.35元
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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. Coverage includes the construction of the Spin groups, Schur Orthogonality, the Peter-Weyl Theorem, the Plancherel Theorem, the Weyl Integration and Character Formulas, the Highest Weight Classification, and the Borel-Weil Theorem. the book develops the necessary Lie algebra theory with a streamlined approach focusing on linear Lie groups.
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目录


  • 1 Compact Lie Groups

    1.1 Basic Notions

    1.1.1 Manifolds

    1.1.2 Lie Groups

    1.1.3 Lie Subgroups and Homomorphisms

    1.1.4 Compact Classical Lie Groups

    1.1.5 Exercises

    1.2 Basic Topology

    1.2.1 Connectedness

    1.2.2 Simply Connected Cover

    1.2.3 Exercises

    1.3 The Double Cover of SO(n)

    1.3.1 Clifford Algebras

    1.3.2 Spinn(R)and Pinn(R)

    1.3.3 Exercises

    1.4 Integration

    1.4.1 Volume Forms

    1.4.2 Invariant Integration

    1.4.3 Fubini's Theorem

    1.4.4 Exercises

    2 Representations

    2.1 Basic Notions

    2.1.1 Definitions

    2.1.2 Examples

    2.1.3 Exercises

    2.2 Operations on Representations

    2.2.1 Constructing New Representations

    2.2.2 Irreducibility and Schur's Lemma

    2.2.3 Unitarity

    2.2.4 Canonical Decomposition

    2.2.5 Exercises

    2.3 Examples of Irreducibility

    2.3.1 SU(2)and Vn(C2)

    2.3.2 SO(n)and Harmonic Polynomials

    2.3.3 Spin and Half-Spin Representations

    2.3.4 Exercises

    3 Harmonic Analysis

    3.1 Matrix Coefficients

    3.1.1 Schur Orthogonality

    3.1.2 Characters

    3.1.3 Exercises

    3.2 infinite-Dimensional Representations

    3.2.1 Basic Definitions and Schur's Lemma

    3.2.2 G-Finite Vectors

    3.2.3 Canonical Decomposition

    3.2.4 Exercises

    3.3 The Peter-Weyl Theorem

    3.3.1 The Left and Right Regular Representation

    3.3.2 Main Result

    3.3.3 Applications

    3.3.4 Exercises

    3.4 Fourier Theory

    3.4.1 Convolution

    3.4.2 Plancherel Theorem

    3.4.3 Projection Operators and More General Spaces

    3.4.4 Exercises

    4 Lie Algebras

    4.1 Basic Definitions

    4.1.1 Lie Algebras of Linear Lie Groups

    4.1.2 Exponential Map

    4.1.3 Lie Algebras for the Compact Classical Lie Groups

    4.1.4 Exercises

    4.2 Further Constructions

    4.2.1 Lie Algebra Homomorphisms

    4.2.2 Lie Subgroups and Subalgebras

    4.2.3 Covering Homomorphisms

    4.2.4 Exercises

    5 Abelian Lie Subgroups and Structure

    5.1 Abelian Subgroups and Subalgebras

    5.1.1 Maximal Tori and Caftan Subalgebras

    5.1.2 Examples

    5.1.3 Conjugacy of Cartan Subalgebras

    5.1.4 Maximal Torus Theorem

    5.1.5 Exercises

    5.2 Structure

    5.2.1 Exponential Map Revisited

    5.2.2 Lie Algebra Structure

    5.2.3 Commutator Theorem

    5.2.4 Compact Lie Group Structure

    5.2.5 Exercises

    6 Roots and Associated Structures

    6.1 Root Theory

    6.1.1 Representations of Lie Algebras

    6.1.2 Complexification of Lie Algebras

    6.1.3 Weights

    6.1.4 Roots

    6.1.5 Compact Classical Lie Group Examples

    6.1.6 Exercises

    6.2 The Standard sl(2,C)Triple

    6.2.1 Cartan Involution

    6.2.2 Killing Form

    6.2.3 The Standard sl(2,C)and su(2)Triples

    6.2.4 Exercises

    6.3 Lattices

    6.3.1 Definitions

    6.3.2 Relations

    6.3.3 Center and Fundamental Group

    6.3.4 Exercises

    6.4 Weyl Group

    6.4.1 Group Picture

    6.4.2 Classical Examples

    6.4.3 Simple Roots and Weyl Chambers

    6.4.4 The Weyl Group as a Reflection Group

    6.4.5 Exercises

    7 Highest Weight Theory

    7.1 Highest Weights

    7.1.1 Exercises

    7.2 Weyl Integration Formula

    7.2.1 Regular Elements

    7.2.2 Main Theorem

    7.2.3 Exercises

    7.3 Weyl Character Formula

    7.3.1 Machinery

    7.3.2 Main Theorem

    7.3.3 Weyl Denominator Formula

    7.3.4 Weyl Dimension Formula

    7.3.5 Highest Weight Classification

    7.3.6 Fundamental Group

    7.3.7 Exercises

    7.4 Borel-Weil Theorem

    7.4.1 Induced Representations

    7.4.2 Complex Structure on G/T

    7.4.3 Holomorphic Functions

    7.4.4 Main Theorem

    7.4.5 Exercises

    References

    Index]]>
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