Compact Lie groups / Mark R. Sepanski.
Series Graduate texts in mathematics ; 235Editor: New York, N.Y. : Springer, c2007Descripción: xii, 198 p. : ill. ; 25 cmISBN: 0387302638; 9780387302638Tema(s): Lie groups | Compact groupsOtra clasificación: 22Exx Recursos en línea: Table of contents only | Publisher descriptionPreface xi -- 1 Compact Lie Groups [1] -- 1.1 Basic Notions [1] -- 1.1.1 Manifolds [1] -- 1.1.2 Lie Groups [2] -- 1.1.3 Lie Subgroups and Homomorphisms [2] -- 1.1.4 Compact Classical Lie Groups [4] -- 1.1.5 Exercises [6] -- 1.2 Basic Topology [8] -- 1.2.1 Connectedness [8] -- 1.2.2 Simply Connected Cover [10] -- 1.2.3 Exercises [12] -- 1.3 The Double Cover of SO(n) [13] -- 1.3.1 Clifford Algebras [14] -- 1.3.2 Spin, (R) and Pin„(R) [16] -- 1.3.3 Exercises [18] -- 1.4 Integration [19] -- 1.4.1 Volume Forms [19] -- 1.4.2 Invariant Integration [20] -- 1.4.3 Fubini’s Theorem [22] -- 1.4.4 Exercises [23] -- 2 Representations [27] -- 2.1 Basic Notions [27] -- 2.1.1 Definitions [27] -- 2.1.2 Examples [28] -- 2.1.3 Exercises [32] -- 2.2 Operations on Representations [34] -- 2.2.1 Constructing New Representations [34] -- 2.2.2 Irreducibility and Schur´s Lemma [36] -- 2.2.3 Unitarity [37] -- 2 2.4 Canonical Decomposition [39] -- 2.2.5 Exercises [40] -- 2.3 Examples of Irreducibility [41] -- 2.3.1 SU(2) and V(C) [41] -- 2.3.2 SO(n)and Harmonic Polynomials [42] -- 2.3.3 Spin and Half-Spin Representations [44] -- 2.3.4 Exercises [45] -- 3 Harmonic Analysis [47] -- 3.1 Matrix Coefficients [47] -- 3.1.1 Schur Orthogonality [47] -- 3.1.2 Characters [47] -- 3.1.3 Exercises [49] -- 3.2 Infinite-Dimensional Representations [52] -- 3.2.1 Basic Definitions and Schur's Lemma [54] -- 3.2.2 G-Fimte Vectors [54] -- 3.2.3 Canonical Decomposition [56] -- 3.2.4 Exercises [59] -- 3.3 The Peter-Weyl Theorem [60] -- 3.3.1 The Left and Right Regular Representation [60] -- 3.3.2 Main Result [64] -- 3.3.3 Applications [66] -- 3.3.4 Exercises [69] -- 3.4 Fourier Theory [70] -- 3.4.1 Convolution [71] -- 3.4.2 Plancherel Theorem [72] -- 3.4.3 Projection Operators and More General Spaces [77] -- 3.4.4 Exercises [79] -- 4 Lie Algebras [81] -- 4.1 Basic Definitions [81] -- 4.1.1 Lie Algebras of Linear Lie Groups [81] -- 4.1.2 Exponential Map [83] -- 4.1.3 Lie Algebras for the Compact Classical Lie Groups [84] -- 4.1.4 Exercises [86] -- 4.2 Further Constructions [88] -- 4.2.1 Lie Algebra Homomorphisms [88] -- 4.2.2 Lie Subgroups and Subalgebras [91] -- 4.2.3 Covering Homomorphisms [92] -- 4.2.4 Exercises [93] -- 5.1 Abelian Subgroups and Subalgebras [97] -- 5.1.1 Maximal Tori and Cartan Subalgebras [97] -- 5.1.2 Examples [98] -- 5.1.3 Conjugacy of Cartan Subalgebras [100] -- 5.1.4 Maximal Torus Theorem [102] -- 5.1.5 Exercises [103] -- 5.2 Structure [105] -- 5.2.1 Exponential Map Revisited [105] -- 5.2.2 Lie Algebra Structure [108] -- 5.2.3 Commutator Theorem [109] -- 5.2.4 Compact Lie Group Structure [110] -- 5.2.5 Exercises [111] -- 6 Roots and Associated Structures -- 6.1 Root Theory [113] -- 6.1.1 Rcpfcscnuiions of Lie Algebras [113] -- 6.1.2 Complex trtcotion of Lie Algebras [113] -- 6.1.3 Weights [115] -- 6.1.4 Roots [116] -- 6.1.5 Compact ClassicaL Lie Group Examples [117] -- 6.1.6 Exercises [118] -- 6.2 The Standards1(2, C) Triple [120] -- 6.2.1 Cartin Involution [123] -- 6.2.2 Killing Form [124] -- 6.2.3 The Standard s(2.0 and 2)Triples [125] -- 6.2.4 Exercises [129] -- 6.3 Lattices [130] -- 63.1 Definitions [130] -- 63.2 Relations131 -- 63.3 Center and Fundamental Group [132] -- 6.3.4 Exercises134 -- 6.4 Weight Group [136] -- 6.4.1 Group Picture [136] -- 6.4.2 Classical Examples137 -- 6.4.3 Simple Routs and Weyl Chambers139 -- 6.4.4 The 1 Group ax a Reflection Group [143] -- 6.4.5 Exercises [146] -- 7 Highest Weight Threory [151] -- 7.1 Highest Weights [151] -- 7.1.1 Exercises [154] -- 7.2 Weyl Integration Formula [156] -- 7.2.1 Regular Elements [156] -- 7.2.2 Main Theorem [159] -- 7.2.3 Exercises [162] -- 7.3 Weyl Character Formula [163] -- 7.3.1 Machinery [163] -- 7.3.2 Main Theorem [166] -- 7.3.3 Weyl Denominator Formula [168] -- 7.3.4 Weyl Dimension Formula [168] -- 7.3.5 Highest Weight Classification [169] -- 7.3.6 Fundamental Group [170] -- 7.3.7 Exercises [173] -- 7.4 Borel-Weil Theorem [176] -- 7.4.1 Induced Representations [176] -- 7.4.2 Complex Structure on G/T [178] -- 7.4.3 Holomorphic Functions [180] -- 7.4.4 Main Theorem [182] -- 7.4.5 Exercises [184] -- References [187] -- Index [193] --
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode |
|---|---|---|---|---|---|---|---|
Libros
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Instituto de Matemática, CONICET-UNS | Últimas adquisiciones | 22 Se479 (Browse shelf) | Available | A-9384 |
Includes bibliographical references (p. [187]-191) and index.
Preface xi --
1 Compact Lie Groups [1] --
1.1 Basic Notions [1] --
1.1.1 Manifolds [1] --
1.1.2 Lie Groups [2] --
1.1.3 Lie Subgroups and Homomorphisms [2] --
1.1.4 Compact Classical Lie Groups [4] --
1.1.5 Exercises [6] --
1.2 Basic Topology [8] --
1.2.1 Connectedness [8] --
1.2.2 Simply Connected Cover [10] --
1.2.3 Exercises [12] --
1.3 The Double Cover of SO(n) [13] --
1.3.1 Clifford Algebras [14] --
1.3.2 Spin, (R) and Pin„(R) [16] --
1.3.3 Exercises [18] --
1.4 Integration [19] --
1.4.1 Volume Forms [19] --
1.4.2 Invariant Integration [20] --
1.4.3 Fubini’s Theorem [22] --
1.4.4 Exercises [23] --
2 Representations [27] --
2.1 Basic Notions [27] --
2.1.1 Definitions [27] --
2.1.2 Examples [28] --
2.1.3 Exercises [32] --
2.2 Operations on Representations [34] --
2.2.1 Constructing New Representations [34] --
2.2.2 Irreducibility and Schur´s Lemma [36] --
2.2.3 Unitarity [37] --
2 2.4 Canonical Decomposition [39] --
2.2.5 Exercises [40] --
2.3 Examples of Irreducibility [41] --
2.3.1 SU(2) and V(C) [41] --
2.3.2 SO(n)and Harmonic Polynomials [42] --
2.3.3 Spin and Half-Spin Representations [44] --
2.3.4 Exercises [45] --
3 Harmonic Analysis [47] --
3.1 Matrix Coefficients [47] --
3.1.1 Schur Orthogonality [47] --
3.1.2 Characters [47] --
3.1.3 Exercises [49] --
3.2 Infinite-Dimensional Representations [52] --
3.2.1 Basic Definitions and Schur's Lemma [54] --
3.2.2 G-Fimte Vectors [54] --
3.2.3 Canonical Decomposition [56] --
3.2.4 Exercises [59] --
3.3 The Peter-Weyl Theorem [60] --
3.3.1 The Left and Right Regular Representation [60] --
3.3.2 Main Result [64] --
3.3.3 Applications [66] --
3.3.4 Exercises [69] --
3.4 Fourier Theory [70] --
3.4.1 Convolution [71] --
3.4.2 Plancherel Theorem [72] --
3.4.3 Projection Operators and More General Spaces [77] --
3.4.4 Exercises [79] --
4 Lie Algebras [81] --
4.1 Basic Definitions [81] --
4.1.1 Lie Algebras of Linear Lie Groups [81] --
4.1.2 Exponential Map [83] --
4.1.3 Lie Algebras for the Compact Classical Lie Groups [84] --
4.1.4 Exercises [86] --
4.2 Further Constructions [88] --
4.2.1 Lie Algebra Homomorphisms [88] --
4.2.2 Lie Subgroups and Subalgebras [91] --
4.2.3 Covering Homomorphisms [92] --
4.2.4 Exercises [93] --
5.1 Abelian Subgroups and Subalgebras [97] --
5.1.1 Maximal Tori and Cartan Subalgebras [97] --
5.1.2 Examples [98] --
5.1.3 Conjugacy of Cartan Subalgebras [100] --
5.1.4 Maximal Torus Theorem [102] --
5.1.5 Exercises [103] --
5.2 Structure [105] --
5.2.1 Exponential Map Revisited [105] --
5.2.2 Lie Algebra Structure [108] --
5.2.3 Commutator Theorem [109] --
5.2.4 Compact Lie Group Structure [110] --
5.2.5 Exercises [111] --
6 Roots and Associated Structures --
6.1 Root Theory [113] --
6.1.1 Rcpfcscnuiions of Lie Algebras [113] --
6.1.2 Complex trtcotion of Lie Algebras [113] --
6.1.3 Weights [115] --
6.1.4 Roots [116] --
6.1.5 Compact ClassicaL Lie Group Examples [117] --
6.1.6 Exercises [118] --
6.2 The Standards1(2, C) Triple [120] --
6.2.1 Cartin Involution [123] --
6.2.2 Killing Form [124] --
6.2.3 The Standard s(2.0 and 2)Triples [125] --
6.2.4 Exercises [129] --
6.3 Lattices [130] --
63.1 Definitions [130] --
63.2 Relations131 --
63.3 Center and Fundamental Group [132] --
6.3.4 Exercises134 --
6.4 Weight Group [136] --
6.4.1 Group Picture [136] --
6.4.2 Classical Examples137 --
6.4.3 Simple Routs and Weyl Chambers139 --
6.4.4 The 1 Group ax a Reflection Group [143] --
6.4.5 Exercises [146] --
7 Highest Weight Threory [151] --
7.1 Highest Weights [151] --
7.1.1 Exercises [154] --
7.2 Weyl Integration Formula [156] --
7.2.1 Regular Elements [156] --
7.2.2 Main Theorem [159] --
7.2.3 Exercises [162] --
7.3 Weyl Character Formula [163] --
7.3.1 Machinery [163] --
7.3.2 Main Theorem [166] --
7.3.3 Weyl Denominator Formula [168] --
7.3.4 Weyl Dimension Formula [168] --
7.3.5 Highest Weight Classification [169] --
7.3.6 Fundamental Group [170] --
7.3.7 Exercises [173] --
7.4 Borel-Weil Theorem [176] --
7.4.1 Induced Representations [176] --
7.4.2 Complex Structure on G/T [178] --
7.4.3 Holomorphic Functions [180] --
7.4.4 Main Theorem [182] --
7.4.5 Exercises [184] --
References [187] --
Index [193] --
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