Goodman / Wallach Symmetry, Representations, and Invariants

1. Auflage 2009
ISBN: 978-0-387-79852-3
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 255, 716 Seiten

Reihe: Graduate Texts in Mathematics

Symmetry is a key ingredient in many mathematical, physical, and biological theories. Using representation theory and invariant theory to analyze the symmetries that arise from group actions, and with strong emphasis on the geometry and basic theory of Lie groups and Lie algebras, Symmetry, Representations, and Invariants is a significant reworking of an earlier highly-acclaimed work by the authors. The result is a comprehensive introduction to Lie theory, representation theory, invariant theory, and algebraic groups, in a new presentation that is more accessible to students and includes a broader range of applications. The philosophy of the earlier book is retained, i.e., presenting the principal theorems of representation theory for the classical matrix groups as motivation for the general theory of reductive groups. The wealth of examples and discussion prepares the reader for the complete arguments now given in the general case. Key Features of Symmetry, Representations, and Invariants: (1) Early chapters suitable for honors undergraduate or beginning graduate courses, requiring only linear algebra, basic abstract algebra, and advanced calculus; (2) Applications to geometry (curvature tensors), topology (Jones polynomial via symmetry), and combinatorics (symmetric group and Young tableaux); (3) Self-contained chapters, appendices, comprehensive bibliography; (4) More than 350 exercises (most with detailed hints for solutions) further explore main concepts; (5) Serves as an excellent main text for a one-year course in Lie group theory; (6) Benefits physicists as well as mathematicians as a reference work.

Dr. Roe Goodman has been a professor for 45 years, and is currently a professor at Rutgers University. He as travelled internationally as a visiting professor to numerous prestigious universities. He has authored two books, and co-authored the previous highly successful version of this book. He has edited two books, and has published over 30 articles in refereed journals.Dr. Nolan R. Wallach has been a professor since 1966, and is currently a professor at the University of California, San Diego. He has authored or co-authored over 100 publications. In 1992, he was the Chair of the Editorial Boards Committee of the American Mathematical Society. He has been an editor of Birkhäuser's series, Mathematics: Theory and Applications, since 2001. In addition to numerous other prizes, recognitions and professional organization affiliations, in 2004 he became and Elected Fellow of the American Academy of Arts and Sciences. 

Autoren/Hrsg.

Goodman, Roe

Wallach, Nolan R.

Inhaltsverzeichnis

1;Contents;5
2;Preface;15
3;Organization and Notation;19
4;Lie Groups and Algebraic Groups;21
4.1;1.1 The Classical Groups;21
4.2;1.2 The Classical Lie Algebras;33
4.3;1.3 Closed Subgroups of GL(n;38
4.4;1.4 Linear Algebraic Groups;55
4.5;1.5 Rational Representations;67
4.6;1.6 Jordan Decomposition;75
4.7;1.7 Real Forms of Complex Algebraic Groups;82
4.8;1.8 Notes;88
5;Structure of Classical Groups;89
5.1;2.1 Semisimple Elements;89
5.2;2.2 Unipotent Elements;97
5.3;2.3 Regular Representations of SL(2;103
5.4;2.4 The Adjoint Representation;111
5.5;2.5 Semisimple Lie Algebras;128
5.6;2.6 Notes;145
6;Highest-Weight Theory;147
6.1;3.1 Roots andWeights;147
6.2;3.2 Irreducible Representations;167
6.3;3.3 Reductivity of Classical Groups;183
6.4;3.4 Notes;194
7;Algebras and Representations;195
7.1;4.1 Representations of Associative Algebras;195
7.2;4.2 Duality for Group Representations;215
7.3;4.3 Group Algebras of Finite Groups;226
7.4;4.4 Representations of Finite Groups;235
7.5;4.5 Notes;244
8;Classical Invariant Theory;245
8.1;5.1 Polynomial Invariants for Reductive Groups;246
8.2;5.2 Polynomial Invariants;257
8.3;5.3 Tensor Invariants;266
8.4;5.4 Polynomial FFT for Classical Groups;276
8.5;5.5 Irreducible Representations of Classical Groups;279
8.6;5.6 Invariant Theory and Duality;298
8.7;5.7 Further Applications of Invariant Theory;313
8.8;5.8 Notes;318
9;Spinors;321
9.1;6.1 Clifford Algebras;321
9.2;6.2 Spin Representations of Orthogonal Lie Algebras;331
9.3;6.3 Spin Groups;336
9.4;6.4 Real Forms of Spin(n;343
9.5;6.5 Notes;347
10;Character Formulas;349
10.1;7.1 Character and Dimension Formulas;349
10.2;7.2 Algebraic Group Approach to the Character Formula;362
10.3;7.3 Compact Group Approach to the Character Formula;374
10.4;7.4 Notes;382
11;Branching Laws;383
11.1;8.1 Branching for Classical Groups;383
11.2;8.2 Branching Laws from Weyl Character Formula;390
11.3;8.3 Proofs of Classical Branching Laws;393
11.4;1);393
11.5;8.4 Notes;404
12;Tensor Representations of GL(V);406
12.1;9.1 Schur–Weyl Duality;406
12.2;9.2 Dual Reductive Pairs;418
12.3;9.3 Young Symmetrizers and Weyl Modules;426
12.4;9.4 Notes;442
13;Tensor Representations of O(V) and Sp(V);444
13.1;10.1 Commuting Algebras on Tensor Spaces;444
13.2;10.2 Decomposition of Harmonic Tensors;454
13.3;10.3 Riemannian Curvature Tensors;470
13.4;10.4 Invariant Theory and Knot Polynomials;480
13.5;10.5 Notes;495
14;Algebraic Groups and Homogeneous Spaces;497
14.1;11.1 General Properties of Linear Algebraic Groups;497
14.2;11.2 Structure of Algebraic Groups;509
14.3;11.3 Homogeneous Spaces;518
14.4;11.4 Borel Subgroups;537
14.5;11.5 Further Properties of Real Forms;549
14.6;11.6 Gauss Decomposition;556
14.7;11.7 Notes;561
15;Representations on Spaces of Regular Functions;563
15.1;12.1 Some General Results;563
15.2;12.2 Multiplicity-Free Spaces;569
15.3;1);572
15.4;12.3 Regular Functions on Symmetric Spaces;584
15.5;12.4 Isotropy Representations of Symmetric Spaces;606
15.6;12.5 Notes;627
16;Algebraic Geometry;629
16.1;A.1 Affine Algebraic Sets;629
16.2;A.2 Maps of Algebraic Sets;640
16.3;A.3 Tangent Spaces;646
16.4;A.4 Projective and Quasiprojective Sets;653
17;Linear and Multilinear Algebra;660
17.1;B.1 Jordan Decomposition;660
17.2;B.2 Multilinear Algebra;662
18;Associative Algebras and Lie Algebras;678
18.1;C.1 Some Associative Algebras;678
18.2;C.2 Universal Enveloping Algebras;685
19;Manifolds and Lie Groups;691
19.1;D.1 C;691
19.2;Manifolds;691
19.3;D.2 Lie Groups;703
19.4;References;713
20;Index of Symbols;720
21;Subject Index;724