Algebraic Groups

Definition Additive Group Scheme 𝔾_a admitted

Topic Adjoint Action admitted

Topic Admissible Sets admitted

Topic Affine Algebraic Groups admitted

Theorem Affine Finite-Type Groups Have Faithful Finite-Dimensional Representations admitted

Theorem Affine G-Varieties Embed Equivariantly in Representations admitted

Definition Affine Group Scheme over a Base admitted

Topic Affine Root Systems admitted

Topic Affine Weyl Groups admitted

Theorem Algebra Homomorphisms Form a Group Under Convolution admitted

Topic Algebraic Fundamental Group admitted

Definition Algebraic Group Actions, Orbits, and Stabilizers admitted

Definition Algebraic Group over a Base admitted

Theorem Algebraic Groups Are Closed Subgroups of GL_n admitted

Topic Algebraic Tori admitted

Definition Algebraic Torus admitted

Definition Anisotropic Torus admitted

Theorem Antipode of a Commutative Hopf Algebra Is Multiplicative admitted

Theorem Antipode of a Hopf Algebra Is Anti-Multiplicative admitted

Topic Automorphism Groups admitted

Topic Automorphisms of Root Data admitted

Topic B(G) admitted

Topic BN-Pairs and Tits Systems admitted

Topic Bala-Carter Classification admitted

Topic Base Change admitted

Theorem Base Change Preserves Algebraic Group Structure admitted

Topic Based Root Data admitted

Topic Basic Examples admitted

Topic Basic Sigma-Conjugacy Classes admitted

Theorem Borel Fixed Point Theorem admitted

Topic Borel Subgroups admitted

Theorem Borel Subgroups Are Conjugate and Control Parabolics admitted

Theorem Bruhat Decomposition admitted

Theorem Bruhat Order Gives Schubert Closure Relations admitted

Topic Bruhat-Tits Apartments admitted

Topic Bruhat-Tits Buildings admitted

Topic Brylinski-Deligne K2-Extensions admitted

Topic Cartan Matrices and Dynkin Diagrams admitted

Theorem Cartan Subgroups from Maximal Tori admitted

Topic Centers and Dual Centers admitted

Topic Central Extensions of Algebraic Groups admitted

Topic Central Isogenies admitted

Topic Central Isogenies and Duality admitted

Topic Character and Cocharacter Lattices admitted

Theorem Characterization of Diagonalizable Groups admitted

Theorem Characters Are Group-Like Elements admitted

Definition Chevalley Basis admitted

Theorem Chevalley Existence Theorem admitted

Definition Chevalley Group Construction from a Root Datum admitted

Theorem Chevalley Isomorphism Theorem admitted

Theorem Chevalley Isomorphism Theorem for F-Split Groups admitted

Topic Classical Groups admitted

Theorem Classification of Irreducible Root Systems admitted

Theorem Classification of Rank-One Tori admitted

Example Classification of Real Tori admitted

Topic Classification of Split Reductive Groups admitted

Topic Cocycles and Twisting admitted

Theorem Commutative Algebraic Groups Split into Semisimple and Unipotent Parts admitted

Definition Comodule over a Coalgebra admitted

Theorem Composition with the Antipode Is an Algebra Hom for Commutative Hopf Algebras admitted

Topic Conjugacy Classes admitted

Definition Conjugacy Classes in the Weyl Group admitted

Topic Connected Components admitted

Theorem Connected Subgroups Generate a Closed Connected Subgroup admitted

Theorem Convolution Is Pointwise Multiplication admitted

Definition Convolution of Algebra Homomorphisms admitted

Theorem Convolution of Algebra Homomorphisms Is an Algebra Homomorphism admitted

Definition Coordinate Hopf Algebra of an Affine Group Scheme admitted

Topic Coordinate Rings and Hopf Algebras admitted

Topic Coxeter Complexes admitted

Topic Coxeter Systems and Bruhat Order admitted

Definition Crystallographic and Reduced Root Systems admitted

Definition Definition of Linear Algebraic Groups needs_definition_review

Topic Derived Group, Center, Adjoint Group, and Simply Connected Cover admitted

Topic Diagonalizable Groups admitted

Theorem Diagonalizable Groups Are Dual to Character Groups admitted

Topic Dual Groups for Covers admitted

Topic Dual Reductive Groups admitted

Topic Dual Root Data admitted

Topic Dual Root Systems admitted

Theorem Embedding Theorem for Algebraic Groups admitted

Theorem Essential Image of Spec Is Affine Group Schemes admitted

Topic Exceptional Groups admitted

Topic Extended Groups for Real Forms admitted

Theorem F-Tori Are Classified by Galois Modules admitted

Topic Facets in Bruhat-Tits Buildings admitted

Definition Faithful Linear Representation admitted

Theorem Field of Definition for Subgroups of F-Groups admitted

Lemma Finite-Dimensional Subspace in a Sub-Comodule admitted

Theorem Finiteness of Tori of Bounded Rank admitted

Theorem Fixed Point Theorem for F-Split Solvable Groups admitted

Topic Forms of Algebraic Groups admitted

Topic Forms of Algebraic Objects admitted

Definition Functor of Points of a Group Scheme admitted

Topic Fundamental Group of a Root Datum admitted

Theorem G-Representations Are A-Comodules admitted

Topic Galois Action on Root Data admitted

Topic Galois Cohomology admitted

Theorem Galois Descent for Hopf Algebras admitted

Theorem Galois Descent for Vector Spaces admitted

Topic Galois Groups and Descent Data admitted

Definition General Linear Group Scheme GL(V) formalized

Definition General Linear Group Scheme GL_n admitted

Definition Geometric Conjugacy and Deligne–Lusztig Theory admitted

Theorem Group Object Structure on Spec A via Yoneda admitted

Theorem Hom Into Spec via Global Sections (Spec ⊣ Γ Adjunction) admitted

Definition Homomorphism of Affine Group Schemes admitted

Theorem Homomorphisms of Tori Correspond to Lattice Maps admitted

Theorem Identity Component Is Closed Normal of Finite Index admitted

Theorem Image of an Algebraic Group Homomorphism Is Closed admitted

Definition Indexed Root Datum of an F-Reductive Group admitted

Definition Induced Torus admitted

Topic Inner Forms admitted

Topic Isocrystals with G-Structure admitted

Definition Isogeny Types from the Root Datum admitted

Topic Jacobson-Morozov Theorem admitted

Theorem Jordan Decomposition in Algebraic Groups admitted

Theorem Jordan Decomposition in a Linear Algebraic Group admitted

Topic Kaletha Refined Inner Forms admitted

Topic Kernels, Images, and Quotients admitted

Topic Kottwitz Homomorphism admitted

Topic Kottwitz Map on B(G) admitted

Topic L-Groups admitted

Topic L-Groups for Covers admitted

Topic L-Morphisms admitted

Theorem Lang Theorem for Connected Algebraic Groups over Finite Fields admitted

Topic Langlands Dual Groups admitted

Theorem Levi Decomposition for Parabolic F-Subgroups admitted

Theorem Levi Decomposition of Parabolic Subgroups admitted

Topic Levi Subgroups and Unipotent Radicals admitted

Definition Lie Algebra of an Algebraic Group admitted

Topic Lie Algebras of Reductive Groups admitted

Theorem Lie-Kolchin Theorem admitted

Topic Linear Algebraic Groups admitted

Topic Local and Global Metaplectic Covers admitted

Theorem Maximal F-Split Tori Are G(F)-Conjugate admitted

Theorem Maximal F-Tori Exist in Connected F-Groups admitted

Topic Maximal Tori admitted

Theorem Maximal Tori Are Conjugate admitted

Topic Morphisms and Closed Subgroups admitted

Topic Moy-Prasad Filtrations admitted

Lemma Multiplication of a Commutative Bialgebra Is a Coalgebra Hom admitted

Definition Multiplicative Group Scheme 𝔾_m admitted

Topic Newton Map admitted

Topic Nilpotent Cone admitted

Topic Nilpotent Elements admitted

Topic Nilpotent Orbits admitted

Topic Nonabelian H1 admitted

Example Norm-One Torus of a Separable Extension admitted

Theorem Orbits of Algebraic Group Actions Are Locally Closed admitted

Topic Outer Forms admitted

Topic Parabolic Subgroups admitted

Topic Parahoric Group Schemes admitted

Topic Parahoric Subgroups admitted

Topic Pinned Dual Groups admitted

Topic Pinnings admitted

Theorem Points of GL_n Are Invertible n×n Matrices over R admitted

Theorem Points of 𝔾_a Are the Additive Group of R admitted

Theorem Points of 𝔾_m Are the Units of R admitted

Topic Pure Inner Twists admitted

Topic Quasi-Split Forms admitted

Theorem Quasi-Split and Anisotropic Groups via the Indexed Root Datum admitted

Theorem Quotient by a Closed Subgroup Exists admitted

Theorem R-Points of a Group Scheme Form a Group admitted

Theorem Radical and Derived Group of a Reductive Group admitted

Definition Rank and Split Rank of a Torus admitted

Definition Rank-One Subgroups and SL₂-Embeddings admitted

Theorem Rational Bruhat Decomposition admitted

Topic Rational Forms of Linear Algebraic Groups admitted

Topic Rational Forms of Reductive Groups admitted

Topic Rational Nilpotent Orbits admitted

Topic Real Forms admitted

Topic Reductive Groups admitted

Topic Reductive Groups over Local Fields admitted

Topic Reductive Quotients of Parahoric Group Schemes admitted

Definition Regular Elements admitted

Example Regular Representation as a Comodule admitted

Theorem Relative Root Datum of an F-Reductive Group admitted

Theorem Relative Weyl Group Is Generated by Relative Reflections admitted

Theorem Resolution of a Torus by Induced Tori admitted

Topic Rigid Inner Forms admitted

Topic Rigid Inner Twists admitted

Theorem Root Chains and the String Property admitted

Definition Root Data admitted

Definition Root Datum of a Reductive Group admitted

Theorem Root Form and Positive Definiteness admitted

Definition Root Pairings (Unified Framework) admitted

Theorem Root Subgroups Generate a Reductive Group admitted

Definition Root Systems admitted

Definition Roots of Unity Group Scheme μ_n admitted

Topic Roots, Coroots, and Root Subgroups admitted

Topic SL2-Triples admitted

Definition Semisimple Conjugacy Classes admitted

Topic Semisimple Groups admitted

Topic Sigma-Conjugacy Classes admitted

Topic Smooth Affine Group Schemes over Valuation Rings admitted

Topic Smoothness, Tangent Spaces, and Lie Algebras admitted

Topic Solvable Groups admitted

Definition Spec Functor on Hopf Algebras admitted

Theorem Spec Functor on Hopf Algebras Is Fully Faithful admitted

Definition Special Linear Group Scheme SL_n admitted

Definition Special Orthogonal Group Scheme SO_n admitted

Topic Spherical Buildings admitted

Definition Split Reductive Groups admitted

Definition Split Torus admitted

Theorem Split-Anisotropic Decomposition of a Torus admitted

Topic Splittings over Integral and Parahoric Subgroups admitted

Theorem Springer Correspondence admitted

Definition Stable Conjugacy Classes admitted

Theorem Standard Parabolics Are Classified by Simple Roots admitted

Topic Strong Inner Forms admitted

Topic Strong Real Forms admitted

Definition Strongly Regular Elements and Harish-Chandra Theory admitted

Theorem Structure of Connected Solvable Algebraic Groups admitted

Theorem Subtori and Quotients via the Character Lattice admitted

Definition Symplectic Group Scheme Sp_{2n} admitted

Definition Tamely Ramified Torus admitted

Definition The Diagonalizable Group Scheme D(M) admitted

Theorem The Minimal Splitting Field of a Torus admitted

Definition The Twisted Form of a Galois Lattice admitted

Topic Tits Buildings admitted

Theorem Tori Are Classified by Cocycles in GL_n(Z) admitted

Theorem Tori among Groups of Multiplicative Type admitted

Example Tori over Finite Fields admitted

Theorem Tori up to Isogeny Are Rational Representations admitted

Topic Torsors admitted

Definition Trivial Group Scheme over a Base admitted

Example Trivial Representation as a Comodule admitted

Topic Twisted Conjugacy Classes admitted

Definition Unipotent Elements and the Unipotent Variety admitted

Topic Unipotent Groups admitted

Definition Unipotent Orbits admitted

Theorem Unipotent Radical of a Borel Subgroup and Positive Roots admitted

Theorem Unipotent Subgroups Are Triangularizable admitted

Definition Unramified Torus admitted

Topic Valued Root Data admitted

Topic Vogan Diagrams admitted

Definition Weakly Induced Torus admitted

Topic Weighted Dynkin Diagrams admitted

Theorem Weil Restriction of Tori admitted

Theorem Weyl Group Is Generated by Root Reflections admitted

Topic Weyl Groups admitted

Theorem Weyl Groups as Coxeter Groups admitted