Chapter VI. Semisimple and Unipotent Elements
FC16 — Humphreys, Linear Algebraic Groups.
This chapter contains 9 definitions, 3 theorems, 5 propositions, 7 lemmas, 1 corollary, 1 construction and 2 other statements.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition: semisimple and nilpotent endomorphisms
An endomorphism is semisimple when its minimal polynomial has distinct roots, equivalently it is diagonalizable over K; nilpotent means some power is zero.
Lemma: additive Jordan–Chevalley decomposition
Every endomorphism x decomposes uniquely as x=x_s+x_n with commuting semisimple and nilpotent parts, both polynomial expressions in x without constant term.
Definition: unipotent endomorphism
An invertible endomorphism is unipotent when it is 1 plus a nilpotent endomorphism, equivalently its only eigenvalue is 1.
Lemma: multiplicative Jordan decomposition in GL(V)
Every x∈GL(V) decomposes uniquely as x=x_sx_u=x_ux_s with semisimple x_s and unipotent x_u; the factors preserve every x-stable subspace and commute functorially with commuting operators.
Lemma: one-parameter subgroup generated by a unipotent element in characteristic zero
In characteristic zero, a nontrivial unipotent x determines exp(a log x):G_a→GL(V), an isomorphism onto the smallest closed subgroup containing x.
Proposition: Jordan decomposition of regular representations of GLn and gln
For GL_n and gl_n, the operators induced on coordinate functions by an element have semisimple/unipotent or semisimple/nilpotent parts induced by its matrix Jordan decomposition.
Result: semisimple and unipotent loci under representations
The polynomial nature of Jordan parts in faithful representations permits the intrinsic definitions of semisimple/unipotent group elements and semisimple/nilpotent Lie elements.
Theorem: Jordan decomposition in algebraic groups and Lie algebras
Every x∈G has unique commuting semisimple and unipotent parts in G; every x∈g has unique commuting semisimple and nilpotent parts in g. Homomorphisms and differentials preserve these decompositions.
Definition: simultaneously diagonalizable and trigonalizable sets
A subset of matrices is diagonalizable/trigonalizable when a single conjugation carries it into the diagonal/upper-triangular matrices.
Proposition: simultaneous triangularization of commuting matrices
Every commuting set of matrices is simultaneously triangularizable; any semisimple subfamily can be simultaneously diagonalized.
Theorem: structure of commutative algebraic groups by Jordan parts
For a commutative algebraic group G, the semisimple and unipotent subsets G_s,G_u are closed subgroups, connected when G is; multiplication gives G_s×G_u≅G and Lie algebras split compatibly.
Definition: character group
X(G)=Hom_alg.grp(G,G_m) is an abelian group under multiplication.
Definition: diagonalizable group (d-group)
A d-group is an algebraic group all of whose rational representations decompose into character weight spaces; equivalently it is isomorphic to a closed subgroup of a diagonal group.
Lemma: linear independence of distinct characters
Distinct homomorphisms from an abstract group to K^* are linearly independent as K-valued functions.
Proposition: closed subgroups and character kernels of d-groups
Closed subgroups of d-groups are d-groups and intersections of character kernels; conversely every d-group is diagonalizable.
Definition: torus
A torus is a connected diagonalizable algebraic group, equivalently a group isomorphic to G_m^r.
Lemma: finite generation of character groups of d-groups
For a d-group G, X(G) is a finitely generated abelian group.
Lemma: character group of a connected group is torsion-free
If G is connected then X(G) has no nonzero torsion.
Lemma: independent characters prescribe torus coordinates
Linearly independent characters χ_1,…,χ_r of a torus jointly surject T→G_m^r, so arbitrary nonzero coordinate values can be prescribed.
Theorem: structure of a d-group
A d-group is a direct product G°×H, where G° is a torus and H is finite of order prime to char K; in particular connected d-groups are tori.
Definition: one-parameter multiplicative subgroups and character–cocharacter pairing
Y(D)=Hom(G_m,D) is the cocharacter group; composition with χ∈X(D) gives the integer pairing ⟨χ,λ⟩, perfect for a torus.
Construction: coordinate algebra of a diagonalizable group from its character group
The coordinate algebra of a d-group is described by the group algebra K[X(G)], relating diagonalizable groups contravariantly to finitely generated abelian groups with allowed torsion.
Result: finite-order elements are dense in a d-group
The elements of finite order prime to the characteristic form a dense subset of a diagonalizable group.
Proposition: rigidity for families of homomorphisms from dense torsion
Under the stated hypotheses on a connected parameter variety V, a morphic family of homomorphisms from a group with dense torsion into a group with finitely many m-torsion points is constant in the parameter.
Corollary: normalizer identity component equals centralizer for a d-group
If D is a diagonalizable subgroup of G, then N_G(D)°=C_G(D)°.
Definition: weights of a d-group action
For a rational D-module V, V=⊕_{χ∈X(D)}V_χ; nonzero χ with V_χ≠0 are weights, with V^D=V_0.
Definition: roots relative to a diagonalizable subgroup
For D≤G, the nonzero weights of Ad D on g are roots Φ(G,D); the zero-weight space is g^D and contains d.
Proposition: generic element of a torus has the same centralizer
If D is a torus stabilizing H, there exists x∈D with C_H(x)=C_H(D) and corresponding Lie centralizers equal.