Algebraic homogeneous spaces and invariant theory
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Bibliographic Information
Algebraic homogeneous spaces and invariant theory
(Lecture notes in mathematics, 1673)
Springer-Verlag, c1997
Available at 88 libraries
  Aomori
  Iwate
  Miyagi
  Akita
  Yamagata
  Fukushima
  Ibaraki
  Tochigi
  Gunma
  Saitama
  Chiba
  Tokyo
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  Niigata
  Toyama
  Ishikawa
  Fukui
  Yamanashi
  Nagano
  Gifu
  Shizuoka
  Aichi
  Mie
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  Kyoto
  Osaka
  Hyogo
  Nara
  Wakayama
  Tottori
  Shimane
  Okayama
  Hiroshima
  Yamaguchi
  Tokushima
  Kagawa
  Ehime
  Kochi
  Fukuoka
  Saga
  Nagasaki
  Kumamoto
  Oita
  Miyazaki
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Note
Includes bibliographical references (p. [138]-145) and index
Description and Table of Contents
Description
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.
Table of Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . Chapter One - Observable Subgroups 1. Stabilizer Subgroups . . . . . . . . . . . . . . . 2. Equivalent Conditions. . . . . . . . . . . . . . . 3. Observable Subgroups of Reductive Groups . . . . . 4. Finite Generation of kAEG/HUE. . . . . . . . . . . . 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. . . . . . . . . . . . . . . . . . . . . . . . . . .
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