Chapter II. Affine Algebraic Groups
FC16 — Humphreys, Linear Algebraic Groups.
This chapter contains 12 definitions, 1 theorem, 7 propositions, 2 lemmas, 3 corollaries, 4 examples and 1 construction.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition: affine algebraic group
An algebraic group is an affine variety G whose multiplication G×G→G and inversion G→G are morphisms.
Definition/comparison: abelian variety
A complete algebraic group is called an abelian variety; the text thereafter reserves “algebraic group” for affine algebraic groups.
Examples: additive and multiplicative groups
G_a is A¹ under addition and G_m=A¹−{0} under multiplication, both affine algebraic groups.
Example/construction: general linear group
GL(n,K) is the principal open det≠0 in matrix affine space, with regular multiplication and inversion.
Examples: classical matrix algebraic groups
SL(n), diagonal, triangular, unipotent triangular, symplectic and orthogonal groups are realized as closed algebraic subgroups of GL(n).
Definition: identity component
G° is the irreducible component of the identity.
Proposition: structure of the identity component
G° is a normal subgroup of finite index; its cosets are exactly the connected/irreducible components, and every closed finite-index subgroup contains G°.
Lemma: product of dense open subsets
If U,V are dense open subsets of an algebraic group G, then G=UV.
Proposition: closure of a subgroup
The closure of an abstract subgroup of an algebraic group is a subgroup; a constructible subgroup is closed.
Corollary: product of normalized closed subgroups
If closed B normalizes closed A, then AB is a closed subgroup.
Definition: morphism of algebraic groups
An algebraic-group morphism is simultaneously a group homomorphism and a morphism of varieties.
Proposition: kernel, image, components and dimension of a homomorphism
For φ:G→G′, Kerφ and Imφ are closed, φ(G°)=φ(G)°, and dim G=dim Kerφ+dim Imφ.
Definition: rational representation
A morphism of algebraic groups G→GL(V) is a rational representation.
Definition: group closure generated by a subset
For M⊆G, the group closure is the smallest closed subgroup containing M.
Proposition: generation by irreducible images
Images of irreducible varieties through e generate a connected closed subgroup, and finitely many signed products already equal that subgroup.
Corollary: connected subgroups generate connected group
A group abstractly generated by closed connected subgroups is connected.
Construction/result: Hopf-algebra structure on K[G]
Multiplication, inversion and identity induce comultiplication K[G]→K[G]⊗K[G], antipode and counit satisfying the Hopf identities.
Definition: group action, orbit, stabilizer and fixed points
An action satisfies associativity and identity; G·x is the orbit, G_x the stabilizer, X^G the fixed set, and G/G_x maps bijectively to the orbit.
Examples: translation and conjugation actions
Left/right translation and conjugation are canonical group actions; centralizers and normalizers arise as stabilizers/fixed-point constructions.
Definition: morphic algebraic-group action
An algebraic group acts morphically on X when the action map G×X→X is a morphism.
Definition: transporter
For Y,Z⊆X, Trans_G(Y,Z)={g∈G | gY⊆Z}; stabilizers, centralizers and normalizers are special cases.
Proposition: closedness of transporters and fixed loci
Transporters into closed sets are closed; stabilizers and centralizers are closed subgroups, and fixed-point sets are closed.
Corollary: normalizers and centralizers are closed
Normalizers and centralizers of closed subgroups, and centralizers of elements, are closed algebraic subgroups.
Definition: equivariant morphism
A morphism φ:X→Y between G-varieties is G-equivariant when φ(gx)=gφ(x).
Proposition: geometry of orbits
Every orbit of an algebraic-group action is smooth and locally closed; its boundary is a union of strictly lower-dimensional orbits, so minimal-dimensional orbits are closed.
Definition: semidirect product
If G acts by automorphisms on N, N⋊G has multiplication (n,g)(n′,g′)=(n(g·n′),gg′); N is normal and every element factors uniquely.
Definition: translation of functions
An action on X induces linear K-algebra automorphisms τ_g of K[X]; for G acting on itself these are left and right translations λ_g,ρ_g.
Lemma: subgroup ideal criterion under right translation
For closed H⊆G with vanishing ideal I, H={x∈G | ρ_x(I)⊆I}.
Proposition: finite-dimensional translation-stable enlargement
Every finite-dimensional subspace of K[X] is contained in a finite-dimensional subspace stable under all translations; stability is characterized by the comorphism landing in K[G]⊗F.
Theorem: linearization of affine algebraic groups
Every affine algebraic group is isomorphic to a closed subgroup of some GL(n,K).