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Borel A. — Linear algebraic groups
Borel A. — Linear algebraic groups

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Название: Linear algebraic groups

Автор: Borel A.


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.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2nd enlarged edition

Год издания: 1991

Количество страниц: 288

Добавлена в каталог: 08.02.2006

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Предметный указатель
$-\varepsilon$-$\tau$-hermitian forms      23.8 23.9
$\Delta$-action      24.5
$\varepsilon$-$\sigma$-hermitian forms      23.8 23.9
action      AG.2.4
Action, closed      1.8
Action, free      1.8
Action, principal      1.8
Additive group $\mathrm{G}_a$      1.6
Adjoint representation Ad      3.5
Admissible scalar product      14.7 21.1
Affine k-algebra      AG.5.2
Affine K-schemes      AG.5.2 AG.5.3 AG.5.4 AG.5.5
Affine k-varieties      AG.12.1
Affine morphism      AG.6.5
Affine space      AG.7.1
Affine variety      AG.5.3 AG.5.4
Algebraic curves      AG.18.5
Algebraic group      1.1
Algebraic Lie algebra      §7
Algebraic transformation space      1.7
Almost direct product of groups      xi
Anisotropic (torus)      8.14
Annihilator      AG.3.1 AG.3.5
Antiautomorphism      23.7
Antihermitian form      23.8
Base change      AG.15.8
Base change for fields      AG.2.1
Basis of a root system      14.7
Birationality      AG.8.2
Boolean algebra homomorphism      AG.1.3
Borel subalgebra      14.16
Borel subgroup      11.1 23.4 24.3
Borel — Weil theorem      24.4
Bruhat decomposition      14.11
Canonical Cartan involution      24.6
Cartan involution      24.6
Cartan subgroup      11.13
Cartan — Malcev theorem      24.6
Categorical quotient      6.16
Cellular decomposition (of G/B)      14.11
centralizer      1.7 23.4 24.6
Character (of an algebraic group)      5.2
Characteristic exponent      AG.2.2
Classification over K      24.1 24.2
Classification Theorem      24.1
Closed immersions      AG.5.6
Closed set of roots      14.7
Closed subvariety      AG.14.4
Combinatorial dimension      AG.1.4 AG.3.2 AG.3.4
Comorphism      xi
Complementary root      8.17
Complete variety      AG.7.4
Completely reducible (representation)      8.19
Complex semi-simple Lie algebra representations      24.3
Conjugacy class      9.1
Conjugate variety      AG.14.3
Connected components      AG.1.2
Constructible set      AG.1.3
Convolution (right or left)      3.4
Cross section      6.13
Defect of Q      23.5
density      AG.1.2
Derived series $\{\mathscr{D}^1\mathrm{G}\}$      2.4
Descending central series $\{\mathscr{C}^1\mathrm{G}\}$      2.4
Diagonal map (of a Hopf algebra)      1.2
Diagonal torus      23.4
Diagonalizable group      8.2
Diagonalizable group split over k      8.2
Diagram      24.1
Diffeomorphism      24.7
Differential      AG.16.1
Differential criteria      AG.2.3
DIMENSION      23.7
Dimension of a variety      AG.9.1 AG.9.2 AG.9.3 AG.10.1
Direct spanning      14.3
Division algebra      23.7
Division algebra with involution      23.7
Dual numbers      AG.16.2
Dynkin diagram      14.7
Endomorphism, nilpotent      4.1
Endomorphism, semi-simple      4.1
Endomorphism, unipotent      4.1
Epimorphism      AG.3.5 AG.12.1
Fibre      AG.10.1 AG.13.2
Field extension      AG.2.1
Flag (rational over k)      15.3
Flag variety      10.3
Frobenius isogeny      16.1 24.1
Frobenius morphism      §16
Full ring of fractions      AG.3.1 AG.3.3
Function field      AG.8.1
Functor of points      AG.13.1
Fundamental highest weights      24.3
Galois actions      AG.§14
Galois actions on k-varieties      AG.14.3
Galois actions on vector spaces      AG.14.1
Galois actions, k-structure defined by      AG.14.2
General linear group $\mathrm{GL}_\mathrm{n}$      1.6
Generic points      AG.13.5
Geometric reductivity      24.4
Grassmannian      10.3
Group closure      2.1
Group, algebraic (defined over k)      1.1
Group, anisotropic over k      20.1
Group, diagonalizable (and split over k)      8.2
Group, isotropic over k      20.1
Group, isotropy      1.7
Group, linear algebraic      1.6
Group, nilpotent      2.4
Group, reductive      11.21
Group, reductive, k-split      18.6
Group, semi-simple      11.21
Group, solvable      2.4
Group, solvable, k-split      15.1
Group, trigonalizable (over k)      4.6
Hopf algebra      1.2
Hypersurfaces      AG.9.2
Ideal      AG.3.2 AG.3.3 AG.3.4
Idempotents      AG.2.5
Image      AG.10.1
Index of a quadratic form      23.5
Integral closure      AG.3.6
Integral extensions      AG.3.6
Involutions      23.7 23.9
Irreducible components      AG.1.1 AG.1.2 AG.1.3 AG.3.4 AG.3.8 AG.6.4
Irreducible root system      14.7
Irreducible, components defined over $k_s$      AG.12.3
Isogeny      16.1 §22 23.6
Isogeny (of diagrams)      24.1
Isogeny, central      22.3 22.11
Isogeny, quasi-central      22.3
Isotropy group      AG.2.5 1.7
Jordan decomposition in an affine group      4.4
Jordan decomposition in the Lie algebra of an affine group      4.4
Jordan decomposition, additive      4.2
Jordan decomposition, multiplicative      4.2
k-algebra      AG.5.3 AG.5.4
k-derivations      AG.§15 AG.16.1
k-forms of G      24.2
k-group      1.1
k-index of a k-form      24.2
k-morphic action      1.7
k-morphism      AG.11.3 AG.14.5 1.1
k-rank      21.1 23.1
K-scheme      AG.5.3 AG.5.4 AG.16.3
k-space      AG.5.1
k-split-diagonalizable group      8.2 23.2
k-split-reductive group      18.6
k-split-solvable group      15.1
k-structures      AG.14.2
k-structures on k-algebras      AG.11.2
k-structures on K-schemes      AG.11.3
k-structures on Vector spaces      AG.11.1
k-varieties      AG.14.3 AG.14.4 AG.14.5
Killing — Cartan classification      24.1
Krull dimension      AG.6.4
Krull dimension of A      AG.3.4
Levi subgroup      13.22
Lie algebra (restricted)      3.1
Lie algebra of an algebraic group      3.3
Lie algebra of an algebraic group, complex semi-simple representations      24.3
Lie — Kolchin theorem      10.5
Linear algebraic group      1.6
Linear algebraic group locally trivial fibration      6.13
Linear representations of semi-simple groups      24.4
Local ring      AG.3.1 AG.3.2
Local ring on a variety      AG.6.4
localization      AG.3.1 AG.15.5
Locally closed sets      AG.1.3
Maximal compact subgroups      24.7
Morphism of algebraic groups      AG.5.1 AG.10.1 AG.10.2 AG.10.3 1.1
Morphism of algebraic groups, dominant      AG.8.2 AG.13.4
Nil radical      AG.3.3 AG
Nilpotent elements      AG.2.1 AG.3.3 AG.5.3
Nilpotent endomorphism      4.1
Nilpotent group      2.4
Noether normalization      AG.3.7
Noetherian spaces      AG.1.2 AG.3.4 AG.3.5 AG.3.7 AG.3.9 AG.5.3
Non-degenerate quadratic form      23.5
Normal varieties      AG.§18
Normalization      AG 8.2
normalizer      1.7
Nullstellensatz      AG.3.8
One-parameter group (multiplicative)      8.6
Open immersion      AG.5.6
Open map      AG 8.4
Opposite Borel subgroup      14.1
Opposite parabolic subgroups      14.20
Opposition involution      24.3
Orthogonal groups      1.6(7) 23.4 23.6 23.9
Orthogonal groups in characteristic two      23.6
p-isogeny      24.1
Parabolic subgroup      11.2
Polynomial rings      AG 5.2 AG
Positive roots      14.7
Presheaves      AG.4.1
Prevariety      AG.5.3
Principal open set      AG.3.4
Products of open subschemes      AG.6.1
Products of varieties      AG.9.3
Projective spaces      AG.7.2
Projective varieties      AG.7.3 AG.7.4
Quadratic forms      23.4
Quadratic forms in characteristic two      23.5
Quasi-coherent modules      AG.5.5
Quasi-compactness      AG.1.2
Quasi-projective variety      AG.7.3
Quotient (of V by G)      6.3
Quotient morphism (over k)      6.1
R-split torus      24.6
Radical      11.21
Rank (of an algebraic group)      12.2
Rational functions      AG.8.1
Rational representation      1.6
Rational varieties      AG 3.7
Rationality questions for representations      24.5
Real reductive groups      24.6
Reduced rings      AG.2.1 AG.3.3
Reduced root system      14.7 24.3
Reductive group      11.21
Reflection      13.13 14.7
Regular element      12.2
Regular element in a Lie algebra      18.1
Regular functions      AG.6.3
Regular torus      13.1
Residue class rings      AG 5.3
restrictions      AG.4.1 AG.4.2 AG.4.3
Ring of fractions      AG.2.5
Root (of G relative to T)      8.17
Root group      23.6
Root outside a subgroup      8.17
Root system      14.7
Semi-direct product      1.11
Semi-simple anisotropic kernel      24.2
Semi-simple element in an affine group      4.5
Semi-simple endomorphism      4.1
Semi-simple group      11.21
Semi-simple rank      13.13
Separable extensions      AG.2.2 AG.2.5
Separable field extensions      AG.15.6
Separable points      AG.13.1 AG.13.2
Separating transcendence basis      AG.2.3 AG.3.7
Sheafification      AG.4.3
Sheaves      AG.4.2 AG.4.3
Simple points      AG.§17
Simple roots      14.8
Singular element      12.2
Singular subspaces      23.5
Smooth varieties      AG.17.1
Solvable group      2.4
Special set of roots      14.5
Stability group      1.7
Stalk      AG.4.1 AG.4.3 AG.5.1
Subgroup, Borel      11.1
Subgroup, Cartan      11.13
Subgroup, Cartan parabolic      11.2
Subschemes defined over k      AG.11.4
Subvarieties      AG.6.3
Subvarieties defined over k      AG.12.2
Support of a module      AG.3.5
Symmetric algebra      AG.16.3
Symplectic basis      23.2
Symplectic form      23.8
Symplectic group $\mathrm{Sp}_{2n}$      1.6 23.3 23.9
Tangent bundle      AG.16.2
Tangent bundle lemma      AG.15.9
Tangent spaces      AG.§16
Tensor products      AG.16.7
Tits system      §23 24.6
Torus      8.5
Torus split over k      8.2
Torus, anisotropic      8.14
Torus, regular, semi-regular, singular      13.1
Translation (right or left)      1.9
Transporter      1.7
Trigonalizable (over k)      4.6
Unipotent element in an affine group      4.5
Unipotent endomorphism      4.1
Unipotent group      4.8
Unipotent radical      11.21
Unique factorization domain      AG.3.9
Unirational varieties      AG.13.7
Unitary groups      23.9
Universal k-derivation      AG.15.1
Varieties      AG.6.2
Weight (of a root system)      24.1
Weight (of a torus)      5.2
Weyl chamber (algebraic group)      13.9
Weyl chamber (root system)      14.7
Weyl group (of an algebraic group)      11.19
Weyl-group (of a root system)      14.7
Weyl-module      24.4
Zariski dense subset      AG.13.5 AG.13.7
Zariski tangent space      AG.16.1
Zariski topology      AG.3.4 AG.6.6 AG.8.2
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