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AuthorBorel, Armand. author
TitleLinear Algebraic Groups [electronic resource] / by Armand Borel
ImprintNew York, NY : Springer New York : Imprint: Springer, 1991
Edition Second Enlarged Edition
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Descript XI, 290 p. online resource


This book is a revised and enlarged edition of "Linear Algebraic Groups", published by W.A. Benjamin in 1969. The text of the first edition has been corrected and revised. Accordingly, this book presents foundational material on algebraic groups, Lie algebras, transformation spaces, and quotient spaces. After establishing these basic topics, the text then turns to solvable groups, general properties of linear algebraic groups and Chevally's structure theory of reductive groups over algebraically closed groundfields. The remainder of the book is devoted to rationality questions over non-algebraically closed fields. This second edition has been expanded to include material on central isogenies and the structure of the group of rational points of an isotropic reductive group. The main prerequisite is some familiarity with algebraic geometry. The main notions and results needed are summarized in a chapter with references and brief proofs


AGโ{128}{148}Background Material From Algebraic Geometry -- ยง1. Some Topological Notions -- ยง2. Some Facts from Field Theory -- ยง3. Some Commutative Algebra -- ยง4. Sheaves -- ยง5. Affine K-Schemes, Prevarieties -- ยง6. Products; Varieties -- ยง7. Projective and Complete Varieties -- ยง8. Rational Functions; Dominant Morphisms -- ยง9. Dimension -- ยง10. Images and Fibres of a Morphism -- ยง11. k-structures on K-Schemes -- ยง12. k-Structures on Varieties -- ยง13. Separable points -- ยง14. Galois Criteria for Rationality -- ยง15. Derivations and Differentials -- ยง16. Tangent Spaces -- ยง17. Simple Points -- ยง18. Normal Varieties -- References -- Iโ{128}{148}General Notions Associated With Algebraic Groups -- ยง1. The Notion of an Algebraic Groups -- ยง2. Group Closure; Solvable and Nilpotent Groups -- ยง3. The Lie Algebra of an Algebraic Group -- ยง4. Jordan Decomposition -- II โ{128}{148} Homogeneous Spaces -- ยง5. Semi-Invariants -- ยง6. Homogeneous Spaces -- ยง7. Algebraic Groups in Characteristic Zero -- III Solvable Groups -- ยง8. Diagonalizable Groups and Tori -- ยง9. Conjugacy Classes and Centralizers of Semi-Simple Elements -- ยง10. Connected Solvable Groups -- IVโ{128}{148}Borel Subgroups; Reductive Groups -- ยง11. Borel Subgroups -- ยง12. Cartan Subgroups; Regular Elements -- ยง13. The Borel Subgroups Containing a Given Torus -- ยง14. Root Systems and Bruhat Decomposition in Reductive Groups -- Vโ{128}{148}Rationality Questions -- ยง15. Split Solvable Groups and Subgroups -- ยง16. Groups over Finite Fields -- ยง17. Quotient of a Group by a Lie Subalgebra -- ยง18. Cartan Subgroups over the Groundfield. Unirationality. Splitting of Reductive Groups -- ยง19. Cartan Subgroups of Solvable Groups -- ยง20. Isotropic Reductive Groups -- ยง21. Relative Root System and Bruhat Decomposition for Isotropic Reductive Groups -- ยง22. Central Isogenies -- ยง23. Examples -- ยง24. Survey of Some Other Topics -- A. Classification -- B. Linear Representations -- C. Real Reductive Groups -- References for Chapters I to V -- Index of Definition -- Index of Notation

Mathematics Topological groups Lie groups Mathematics Topological Groups Lie Groups


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