E-Book, Englisch, Band 225, 454 Seiten, eBook
Bump Lie Groups
Erscheinungsjahr 2013
ISBN: 978-1-4757-4094-3
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark
E-Book, Englisch, Band 225, 454 Seiten, eBook
Reihe: Graduate Texts in Mathematics
ISBN: 978-1-4757-4094-3
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark
This book aims to be a course in Lie groups that can be covered in one year with a group of good graduate students. I have attempted to address a problem that anyone teaching this subject must have, which is that the amount of essential material is too much to cover. One approach to this problem is to emphasize the beautiful representation theory of compact groups, and indeed this book can be used for a course of this type if after Chapter 25 one skips ahead to Part III. But I did not want to omit important topics such as the Bruhat decomposition and the theory of symmetric spaces. For these subjects, compact groups are not sufficient. Part I covers standard general properties of representations of compact groups (including Lie groups and other compact groups, such as finite or p adic ones). These include Schur orthogonality, properties of matrix coefficients and the Peter-Weyl Theorem.
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Research
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Weitere Infos & Material
1 Haar Measure.- 2 Schur Orthogonality.- 3 Compact Operators.- 4 The Peter-Weyl Theorem.- 5 Lie Subgroups of GL(n, ?).- 6 Vector Fields.- 7 Left-Invariant Vector Fields.- 8 The Exponential Map.- 9 Tensors and Universal Properties.- 10 The Universal Enveloping Algebra.- 11 Extension of Scalars.- 12 Representations of sl(2, ?).- 13 The Universal Cover.- 14 The Local Frobenius Theorem.- 15 Tori.- 16 Geodesics and Maximal Tori.- 17 Topological Proof of Cartan’s Theorem.- 18 The Weyl Integration Formula.- 19 The Root System.- 20 Examples of Root Systems.- 21 Abstract Weyl Groups.- 22 The Fundamental Group.- 23 Semisimple Compact Groups.- 24 Highest-Weight Vectors.- 25 The Weyl Character Formula.- 26 Spin.- 27 Complexification.- 28 Coxeter Groups.- 29 The Iwasawa Decomposition.- 30 The Bruhat Decomposition.- 31 Symmetric Spaces.- 32 Relative Root Systems.- 33 Embeddings of Lie Groups.- 34 Mackey Theory.- 35 Characters of GL(n, ?).- 36 Duality between Sk and GL(n, ?).- 37 The Jacobi-Trudi Identity.- 38 Schur Polynomials and GL(n, ?).- 39 Schur Polynomials and Sk.- 40 Random Matrix Theory.- 41 Minors of Toeplitz Matrices.- 42 Branching Formulae and Tableaux.- 43 The Cauchy Identity.- 44 Unitary Branching Rules.- 45 The Involution Model for Sk.- 46 Some Symmetric Algebras.- 47 Gelfand Pairs.- 48 Hecke Algebras.- 49 The Philosophy of Cusp Forms.- 50 Cohomology of Grassmannians.- References.
