Chapter V. Characteristic 0 Theory
FC16 — Humphreys, Linear Algebraic Groups.
This chapter contains 3 definitions, 8 theorems, 3 corollaries, 1 construction, 1 convention and 6 other statements.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Convention: characteristic-zero hypothesis
Throughout Chapter V the base field has characteristic zero; separability of morphisms and homogeneous quotients is therefore available.
Definition: algebraic Lie subalgebra
A Lie subalgebra h⊆g is called algebraic (relative to G) when h=Lie(H) for a closed connected subgroup H≤G.
Theorem: lattice correspondence between connected subgroups and Lie subalgebras
For connected G in characteristic zero, H↦Lie(H) is injective and inclusion-preserving on closed connected subgroups.
Result: Lie algebras of vector and subspace stabilizers
For W⊆V and v∈V, the Lie algebras of their GL(V)-stabilizers are exactly the linear stabilizer algebra g_W and annihilator algebra g_v.
Theorem: group and Lie algebra have the same invariant subspaces and vectors
If H≤GL(V) is closed, Lie(H_W)=h∩g_W and Lie(H_v)=h∩g_v; hence a rational H-module and its Lie algebra stabilize the same subspaces and fix the same vectors.
Corollary: automorphisms and derivations coincide infinitesimally
For a finite-dimensional K-algebra A in characteristic zero, Lie(Aut A)=Der A.
Theorem: normal connected subgroups correspond to ideals
For connected G and closed connected H, H is normal in G iff h is an ideal of g.
Recorded result: Lie normalizers in characteristic zero
The source records that Lie(N_G(H)) equals the Lie-algebra normalizer n_g(h) for connected G and closed connected H.
Theorem: centralizers and centers in characteristic zero
For connected G, Lie(C_G(x))=g^{Ad x}; Ker Ad=Z(G) and Lie(Z(G))=Ker ad=z(g).
Corollary/result: commutativity detected by the Lie algebra
A connected algebraic group in characteristic zero is commutative iff its Lie algebra is commutative.
Definition: semisimple Lie algebra and semisimple algebraic group
A positive-dimensional Lie algebra is semisimple when it has no nonzero commutative ideal; a connected positive-dimensional algebraic group is semisimple when it has no nontrivial closed connected commutative normal subgroup.
Theorem: semisimplicity of group and Lie algebra
For connected G in characteristic zero, G is semisimple iff g is semisimple.
Theorem: adjoint image and derivations of a semisimple group
If G is semisimple, Ad(G)=(Aut g)° and ad(g)=Der g; in particular Z(G) is finite.
Recorded classification input: root-system classification of semisimple Lie algebras
The source records that a semisimple Lie algebra is a direct sum of simple ideals and is classified by a disjoint union of irreducible root systems of types A,B,C,D,E,F,G.
Result: semisimple groups with isomorphic Lie algebras are locally isomorphic
If semisimple G,G′ have isomorphic Lie algebras, then G/Z(G)≅G′/Z(G′).
Recorded comparison: centers and root-system fundamental groups
The source records that the center of a semisimple group is constrained by the weight-lattice/root-lattice quotient, anticipating the fundamental group defined later.
Construction/result: almost-direct decomposition from simple Lie ideals
The simple-ideal decomposition g=⊕g_i yields connected normal almost-simple subgroups G_i which centralize one another, generate G, and meet complementary products only in finite central subgroups.
Theorem/recorded result: Weyl complete reducibility
Every finite-dimensional representation of a semisimple Lie algebra in characteristic zero is completely reducible.
Corollary/result: complete reducibility for semisimple algebraic groups
Every rational representation of a semisimple algebraic group in characteristic zero is completely reducible.
Definition: ring of invariants
For a group G acting rationally on a K-algebra R, R^G denotes the subalgebra of elements fixed by G.
Theorem: finite generation of invariant polynomial rings under complete reducibility
If closed G≤GL(n) acts naturally on R=K[T_1,…,T_n] and all rational G-modules are completely reducible, then R^G is finitely generated.
Recorded consequence: finite generation of affine invariant rings
For a rational action on an affine variety X, the same complete-reducibility hypothesis yields finite generation of K[X]^G.