E-Book, Englisch, Band 1673, 152 Seiten, eBook
Reihe: Lecture Notes in Mathematics
Grosshans Algebraic Homogeneous Spaces and Invariant Theory
1997
ISBN: 978-3-540-69617-9
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark
E-Book, Englisch, Band 1673, 152 Seiten, eBook
Reihe: Lecture Notes in Mathematics
ISBN: 978-3-540-69617-9
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark
The invariant theory of non-reductive groups has its roots in the 19th century but has seen some very interesting developments in the past twenty years. This book is an exposition of several related topics including observable subgroups, induced modules, maximal unipotent subgroups of reductive groups and the method of U-invariants, and the complexity of an action. Much of this material has not appeared previously in book form. The exposition assumes a basic knowledge of algebraic groups and then develops each topic systematically with applications to invariant theory. Exercises are included as well as many examples, some of which are related to geometry and physics.
Zielgruppe
Research
Autoren/Hrsg.
Weitere Infos & Material
Introduction . . . . . . . . . . . . . . . . . . . . . . Chapter One - Observable Subgroups 1. Stabilizer Subgroups . . . . . . . . . . . . . . . 2. Equivalent Conditions. . . . . . . . . . . . . . . 3. Observable Subgroups of Reductive Groups . . . . . 4. Finite Generation of kÄG/HÜ. . . . . . . . . . . . Appendix: On Valuation Rings. . . . . . . . . 5. Maximal Unipotent Subgroups. . . . . . . . . . . . Bibliographical Note. . . . . . . . . . . . . . . . . . Chapter Two - The Transfer Principle 6. Induced Modules. . . . . . . . . . . . . . . . . . Appendix: Affine Quotients and induced modules 7. Induced Modules and Observable Subgroups . . . . . Appendix: On a Theorem of F. A. Bogomolov . . 8. Counter-examples . . . . . . . . . . . . . . . . . 9. The Transfer Principle . . . . . . . . . . . . . . 10. The Theorems of Roberts and Weitzenb'ck. . . . . . 11. Geometric Examples . . . . . . . . . . . . . . . . A. Multiplicity-free actions . . . . . . . . B. Affine Geometry . . . . . . . . . . . . . C. Invariants of the Orthogonal Group. . . . D. Euclidean Geometry. . . . . . . . . . . . E. Hilbert's Example. . . . . . . . . . . . Chapter Three - Invariants of Maximal Unipotent Subgroups 12. The Representations E( ) . . . . . . . . . . . . . 13. An Example: The General Linear Group . . . . . . . A. Straightening . . . . . . . . . . . . . . B. U - invariants. . . . . . . . . . . . . . C. Results of K. Pommerening . . . . . . . . 14. The Relationship between A and G AU. . . . . . . . 15. The Algebra grA. . . . . . . . . . . . . . . . . . 16. Finite Generation and U-invariants . . . . . . . . A. Algebras. . . . . . . . . . . . . . . . . B. Modules . . . . . . . . . . . . . . . . . 17. S-varieties. . . . . . . . . . . . . . . . . . . . 18. Flat Deformations and Normality. . . . . . . . . . Bibliographical Note. . . . . . . . . . . . . . . . . . Chapter Four - Complexity 19. Basic Principles . . . . . . . . . . . . . . . . . Appendix: On Quotient Spaces . . .. . . . . 20. Unique Factorization Domains . . . . . . . . . . . A. c(X) = 0. . . . . . . . . . . . . . . . . B. c(X) = 1. . . . . . . . . . . . . . . . . 21. Complexity and Finite Generation . . . . . . . . . A. Statement of Results. . . . . . . . . . . B. Proof of Theorem 21.1 . . . . . . . . . . 22. Spherical Subgroups. . . . . . . . . . . . . . . . 23. Finite Generation of Induced Modules . . . . . . . A. Condition (FM). . . . . . . . . . . . . . B. Epimorphic Subgroups. . . . . . . . . . . Bibliographical Note. . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . List of Symbols. . . . . . . . . . . . . . . . . . . . . . Index. . . . . . . . . . . . . . . . . . . . . . . . . . .
