Chapter IV. Isomorphism and Conjugacy Theorems
FC14 — Humphreys, Introduction to Lie Algebras and Representation Theory.
This chapter contains 7 definitions, 5 theorems, 3 propositions, 9 lemmas, 5 corollaries, 4 constructions and 2 conventions.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Proposition: simple Lie algebras have irreducible root systems
If L is simple and H is maximal toral, the corresponding root system Φ is irreducible.
Corollary: simple-ideal decomposition matches irreducible root components
If semisimple L=L_1⊕⋯⊕L_t and H is maximal toral, then H_i=H∩L_i is maximal toral in L_i; the root systems Φ_i of (L_i,H_i) are the irreducible components of Φ and Φ=Φ_1∪⋯∪Φ_t.
Proposition: generation by simple root spaces
For semisimple L with maximal toral H and base Δ of Φ, L is generated by L_α and L_{−α} for α∈Δ; equivalently by arbitrary nonzero root vectors x_α∈L_α and y_α∈L_{−α}.
Definition: standard set of generators
After choosing nonzero x_α∈L_α and the unique y_α∈L_{−α} with [x_α,y_α]=h_α for α∈Δ, the sets {x_α,y_α} and {x_α,y_α,h_α} are called standard sets of generators.
Theorem: Theorem 14.2 (isomorphism theorem)
Let simple L,L′ have maximal toral subalgebras H,H′ and isomorphic root systems Φ≅Φ′. The induced H≅H′ together with arbitrary chosen isomorphisms L_α→L_{α′} on simple-root spaces extends uniquely to a Lie-algebra isomorphism L→L′.
Corollary: semisimple Lie algebra determined by its root system
The isomorphism theorem extends componentwise: a semisimple Lie algebra over F is determined up to isomorphism by its root system.
Proposition: opposition involution
For semisimple L with fixed standard generators, there is an automorphism σ of order two satisfying σ(x_α)=−y_α, σ(y_α)=−x_α for simple α and σ(h)=−h for h∈H.
Construction: inner automorphism realizing a root reflection
For α∈Φ, τ_α=exp(ad x_α) exp(ad(−y_α)) exp(ad x_α) is inner, restricts to the reflection σ_α on H, and sends each root space L_β onto L_{σ_αβ}. Hence Weyl-group elements are represented by inner automorphisms, though not canonically as a subgroup.
Convention: field assumptions for Cartan-subalgebra theory
For §15 the field is algebraically closed; its characteristic is arbitrary except where explicitly restricted.
Construction: generalized eigenspace decomposition for ad x
For x∈L, write L=⊕{a∈F}L_a(ad x), where L_a(ad x) is the generalized a-eigenspace; write L=L_0(ad x)⊕L(ad x), with L_ the sum over a≠0. The same notation applies to ad x-stable subalgebras.
Lemma: brackets of generalized ad-eigenspaces
[L_a(ad x),L_b(ad x)]⊆L_{a+b}(ad x). Hence L_0(ad x) is a subalgebra; in characteristic zero, every element of L_a(ad x) for a≠0 is ad-nilpotent.
Definition: Engel subalgebra
For x∈L, the subalgebra L_0(ad x) is called an Engel subalgebra.
Lemma: minimal Engel-subalgebra containment
Let K≤L and choose z∈K with L_0(ad z) minimal among L_0(ad x), x∈K. If K⊆L_0(ad z), then L_0(ad z)⊆L_0(ad x) for every x∈K.
Lemma: Engel-containing subalgebras are self-normalizing
If a subalgebra K contains an Engel subalgebra of L, then N_L(K)=K. In particular every Engel subalgebra is self-normalizing.
Definition: Cartan subalgebra (CSA)
A Cartan subalgebra of L is a nilpotent self-normalizing subalgebra.
Theorem: Cartan subalgebras are precisely minimal Engel subalgebras
A subalgebra H is a Cartan subalgebra iff it is a minimal Engel subalgebra. In particular Cartan subalgebras exist under the standing field assumptions.
Corollary: Cartan subalgebras in semisimple characteristic-zero Lie algebras
For semisimple L in characteristic zero, Cartan subalgebras are exactly maximal toral subalgebras.
Definition/result: regular semisimple element
Every maximal toral subalgebra H of a semisimple characteristic-zero L has the form C_L(s) for some semisimple s∈L; such an s is called regular semisimple.
Lemma: epimorphic image of a Cartan subalgebra
If φ:L→L′ is surjective and H is a CSA of L, then φ(H) is a CSA of L′.
Lemma: lifting a Cartan subalgebra through an epimorphism
If φ:L→L′ is surjective, H′ is a CSA of L′, K=φ^{-1}(H′), and H is a CSA of K, then H is a CSA of L.
Convention: field assumptions for conjugacy theory
Throughout §16, F is algebraically closed of characteristic zero.
Definition: strongly ad-nilpotent element and the group E(L)
x∈L is strongly ad-nilpotent if x lies in L_a(ad y) for some y∈L and nonzero eigenvalue a. Let A(L) be this set and E(L) the subgroup of Int(L) generated by exp(ad x), x∈A(L). E(L) is normal in Aut(L).
Construction: relative group E(L;K)
For K≤L, E(L;K) is generated by exp(ad_L x) for x strongly ad-nilpotent in K; restriction to K gives E(K). Strongly ad-nilpotent elements are functorial under epimorphisms.
Lemma: lifting E-automorphisms through epimorphisms
For a surjection φ:L→L′, every σ′∈E(L′) lifts to some σ∈E(L) with φσ=σ′φ.
Theorem: conjugacy of Cartan subalgebras in solvable Lie algebras
If L is solvable, any two Cartan subalgebras of L are conjugate under E(L).
Definition: Borel subalgebra
A Borel subalgebra of L is a maximal solvable subalgebra.
Lemma: Borel subalgebras are self-normalizing
If B is a Borel subalgebra of L, then N_L(B)=B.
Lemma: Borels and the semisimple quotient
If Rad L≠L, Borel subalgebras of L correspond bijectively to Borel subalgebras of the semisimple quotient L/Rad L; every Borel contains Rad L.
Construction: standard Borel and nilradical from a base
For semisimple L, CSA H, root system Φ, and base Δ, set B(Δ)=H⊕⊕{α>0}L_α and N(Δ)=⊕{α>0}L_α. N(Δ) is nilpotent, [B(Δ),B(Δ)]=N(Δ), and B(Δ) is solvable.
Lemma: standard Borels and their conjugacy
B(Δ) is a Borel subalgebra, called standard relative to H. All standard Borel subalgebras relative to H are conjugate under E(L).
Theorem: conjugacy of Borel subalgebras
All Borel subalgebras of an arbitrary Lie algebra L over the standing field are conjugate under E(L).
Corollary: conjugacy of Cartan subalgebras
All Cartan subalgebras of an arbitrary Lie algebra L are conjugate under E(L).
Definition: rank of a Lie algebra
The rank of an arbitrary Lie algebra L is the dimension of a Cartan subalgebra. This is well-defined by conjugacy; for semisimple L it equals the rank of its root system.
Corollary: Borels containing a fixed Cartan subalgebra are standard
For semisimple L with fixed CSA H, every Borel subalgebra containing H equals B(Δ) for some base Δ of the root system.
Theorem/description: automorphism group of a semisimple Lie algebra
For semisimple L, every automorphism is a product of an inner automorphism and a graph automorphism; diagonal automorphisms are inner. The source records Aut(L)=Int(L)⋊Γ(L) and E(L)=Int(L), with Γ(L) the Dynkin-diagram automorphism group.