Chapter II. Semisimple Lie Algebras
FC14 — Humphreys, Introduction to Lie Algebras and Representation Theory.
This chapter contains 17 definitions, 9 theorems, 6 propositions, 10 lemmas, 9 corollaries, 2 examples, 7 constructions, 2 conventions, 1 formula and 1 criterion.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Convention: characteristic-zero algebraically closed base field
Except where explicitly stated otherwise, F is algebraically closed of characteristic zero throughout the remaining semisimple theory.
Theorem: common eigenvector theorem for solvable linear Lie algebras
If L⊆gl(V) is solvable, V finite-dimensional and nonzero, then V contains a common eigenvector for all elements of L.
Corollary: Lie’s Theorem
A solvable linear Lie algebra L⊆gl(V) stabilizes a complete flag; equivalently it is simultaneously upper triangularizable.
Corollary: ideal flag in a solvable Lie algebra
Every solvable Lie algebra L has a chain of ideals 0=L_0⊂L_1⊂⋯⊂L_n=L with dim L_i=i.
Corollary: nilpotence of the derived algebra of a solvable Lie algebra
If L is solvable and x∈[L,L], then ad_L x is nilpotent; in particular [L,L] is nilpotent.
Definition: semisimple endomorphism
For finite-dimensional V, x∈End(V) is semisimple if its minimal polynomial has distinct roots; over algebraically closed F this is equivalent to diagonalizability.
Proposition: Jordan–Chevalley decomposition
Every x∈End(V) has unique commuting semisimple and nilpotent parts x_s,x_n with x=x_s+x_n. They are polynomials in x without constant term, commute with every endomorphism commuting with x, and preserve the source/target condition x(B)⊆A between invariant subspaces.
Definition: additive Jordan–Chevalley decomposition
The decomposition x=x_s+x_n of the proposition is the additive Jordan–Chevalley (Jordan) decomposition; x_s and x_n are the semisimple and nilpotent parts.
Lemma: Jordan decomposition under the adjoint representation
If x=x_s+x_n in End(V), then ad x=ad x_s+ad x_n is the Jordan decomposition of ad x on End(V).
Lemma: derivation algebras contain Jordan parts
For a finite-dimensional F-algebra A, Der(A) contains the semisimple and nilpotent parts in End(A) of each derivation.
Lemma: trace criterion for nilpotence
Let A⊆B⊆gl(V) and M={x∈gl(V) | [x,B]⊆A}. If x∈M satisfies Tr(xy)=0 for every y∈M, then x is nilpotent.
Formula: trace commutator identity
For endomorphisms x,y,z of a finite-dimensional vector space, Tr([x,y]z)=Tr(x[y,z]).
Theorem: Cartan’s solvability criterion
If L⊆gl(V) and Tr(xy)=0 for all x∈[L,L], y∈L, then L is solvable.
Corollary: Cartan criterion via adjoint traces
If Tr(ad x ad y)=0 for all x∈[L,L], y∈L, then L is solvable.
Definition: Killing form
The Killing form of L is κ(x,y)=Tr(ad x ad y). It is symmetric bilinear and associative: κ([x,y],z)=κ(x,[y,z]).
Lemma: Killing form on an ideal
If I is an ideal of L, the Killing form of I equals the restriction of the Killing form of L to I×I.
Definition: nondegenerate symmetric form and radical
A symmetric bilinear form B is nondegenerate when its radical {x | B(x,y)=0 for all y} is zero. For the Killing form the radical is an ideal.
Example: Killing form of sl(2,F)
For char F≠2, the Killing form of sl(2,F) has the displayed nonsingular matrix in the standard basis, hence is nondegenerate.
Criterion: semisimplicity via absence of abelian ideals
L is semisimple iff it has no nonzero abelian ideals.
Theorem: Cartan criterion for semisimplicity
A Lie algebra L is semisimple iff its Killing form is nondegenerate.
Definition: direct sum of ideals
L is the direct sum of ideals I_1,…,I_t when L=I_1⊕⋯⊕I_t as vector spaces; then [I_i,I_j]=0 for i≠j and brackets are componentwise.
Theorem: decomposition of a semisimple Lie algebra into simple ideals
Every semisimple L decomposes as a direct sum of simple ideals L_1⊕⋯⊕L_t. These are exactly the simple ideals of L, and each Killing form is the restriction of that of L.
Corollary: ideals and quotients of a semisimple Lie algebra
For semisimple L, L=[L,L]; every ideal and homomorphic image is semisimple, and every ideal is a sum of some simple summands.
Theorem: all derivations of a semisimple Lie algebra are inner
If L is semisimple, Der(L)=ad(L).
Definition: abstract Jordan decomposition in a semisimple Lie algebra
For x∈L semisimple, unique s,n∈L satisfy x=s+n, [s,n]=0 and ad x=ad s+ad n is the ordinary Jordan decomposition. These are the ad-semisimple and ad-nilpotent parts of x.
Convention: finite-dimensional representations in §6
All representations in §6 are finite-dimensional unless explicitly stated otherwise.
Definition: L-module
An L-module is a vector space V with a bilinear action satisfying [x,y]v=x(yv)−y(xv); equivalently it is a representation L→gl(V).
Definition: homomorphism and isomorphism of L-modules
An L-module homomorphism is a linear map commuting with the L-action; an isomorphism is a vector-space isomorphism of this kind, and isomorphic modules afford equivalent representations.
Definition: irreducible L-module
A nonzero L-module is irreducible if its only L-submodules are 0 and itself.
Definition: completely reducible L-module
V is completely reducible if it is a direct sum of irreducible L-submodules; equivalently every submodule has an L-stable complement.
Construction: direct sum of L-modules
For L-modules W,W′, the vector-space direct sum becomes an L-module componentwise.
Lemma: Schur’s Lemma
For an irreducible representation φ:L→gl(V), every endomorphism of V commuting with φ(L) is scalar.
Construction: dual/contragredient module
For an L-module V, the dual V* is an L-module by (x·f)(v)=−f(x·v).
Construction: tensor product of L-modules
For L-modules V,W, V⊗W is an L-module via x·(v⊗w)=xv⊗w+v⊗xw.
Construction: Hom and End as L-modules
Hom(V,W)≅V*⊗W carries the action (x·f)(v)=x f(v)−f(xv); in particular End(V) is an L-module.
Definition/proposition: faithful representation and its trace form
A representation is faithful when it is injective. If L is semisimple and φ:L→gl(V) faithful, B(x,y)=Tr(φ(x)φ(y)) is a nondegenerate symmetric associative bilinear form on L.
Definition: Casimir element
For semisimple L, a representation φ, and dual bases x_i,y_i relative to a nondegenerate associative trace form on the faithful part, c_φ=Σφ(x_i)φ(y_i) commutes with φ(L). For faithful irreducible φ it is scalar with trace dim L and value dim L/dim V.
Example: Casimir element for the natural sl(2,F)-module
For the natural two-dimensional representation of sl(2,F), the Casimir element is xy+(1/2)h²+yx=(3/2)I.
Construction: Casimir element for a nonfaithful representation
For nonfaithful φ, decompose semisimple L into simple ideals; restrict to the sum complementary to Kerφ where φ is faithful and perform the Casimir construction there.
Lemma: semisimple representations have trace-zero image
For a representation φ of semisimple L, φ(L)⊆sl(V). Hence L acts trivially on every one-dimensional module.
Theorem: Weyl’s complete reducibility theorem
Every finite-dimensional representation of a semisimple Lie algebra is completely reducible.
Theorem: semisimple linear Lie algebras contain Jordan parts
If L⊆gl(V) is semisimple, it contains the ordinary semisimple and nilpotent parts of every x∈L; hence abstract and ordinary Jordan decompositions coincide.
Corollary: representations preserve Jordan decomposition
For any finite-dimensional representation φ of semisimple L, if x=s+n is the abstract Jordan decomposition, then φ(x)=φ(s)+φ(n) is the ordinary Jordan decomposition.
Definition: weights and weight spaces for sl(2,F)
For an sl(2,F)-module V, V_λ={v | hv=λv}; λ is a weight when V_λ≠0, and V_λ is its weight space.
Lemma: raising and lowering weights
If v∈V_λ, then xv∈V_{λ+2} and yv∈V_{λ−2}.
Definition: maximal vector
A nonzero vector in a weight space V_λ with V_{λ+2}=0 is a maximal vector of weight λ.
Lemma: sl(2) highest-weight basis formulas
For maximal v_0 of weight λ and v_i=(1/i!)y^i v_0: h v_i=(λ−2i)v_i, yv_i=(i+1)v_{i+1}, and xv_i=(λ−i+1)v_{i−1}.
Theorem: classification of finite-dimensional irreducible sl(2,F)-modules
An irreducible sl(2,F)-module of dimension m+1 has weights m,m−2,…,−m, each multiplicity one, a unique maximal vector up to scalar of highest weight m, and the explicit action of the preceding lemma; at most one irreducible module exists in each dimension.
Corollary: weights in arbitrary finite-dimensional sl(2,F)-modules
All h-eigenvalues are integers and occur with their negatives with equal multiplicity; in any irreducible decomposition, the number of summands is dim V_0+dim V_1.
Construction: irreducible module V(m)
For every m≥0, the highest-weight formulas define an irreducible (m+1)-dimensional sl(2,F)-module V(m).
Proposition: Weyl-element symmetry on V(m)
For τ=expφ(x)expφ(−y)expφ(x), τφ(h)τ^{-1}=−φ(h); hence τ exchanges the weight spaces of weights λ and −λ.
Definition: toral and maximal toral subalgebra
A toral subalgebra consists entirely of semisimple elements; it is maximal toral if not properly contained in another toral subalgebra.
Lemma: toral subalgebras are abelian
Every toral subalgebra of a semisimple Lie algebra is abelian.
Definition: root spaces, roots, and root-space decomposition
For maximal toral H, L_α={x | [h,x]=α(h)x for all h∈H}. The nonzero α with L_α≠0 are roots Φ⊆H*. One has L=C_L(H)⊕⊕_{α∈Φ}L_α; the source also calls this root-space decomposition the Cartan decomposition.
Proposition: basic root-space bracket, nilpotence, and orthogonality
[L_α,L_β]⊆L_{α+β}; for α≠0 every x∈L_α is ad-nilpotent; and L_α is Killing-orthogonal to L_β when α+β≠0.
Corollary: Killing form on the zero root space
The restriction of the Killing form to L_0=C_L(H) is nondegenerate.
Lemma: trace of product with a commuting nilpotent
If x,y commute on a finite-dimensional vector space and y is nilpotent, then xy is nilpotent and Tr(xy)=0.
Proposition: centralizer of a maximal toral subalgebra
For maximal toral H in semisimple L, H=C_L(H).
Corollary: Killing form identifies H with H
The Killing form restricts nondegenerately to H, so each λ∈H* has a unique t_λ∈H with λ(h)=κ(t_λ,h).
Proposition: root-pair geometry and sl(2)-triples
Φ spans H*; α∈Φ implies −α∈Φ; [x,y]=κ(x,y)t_α for x∈L_α,y∈L_{−α}; [L_α,L_{−α}]=F t_α; α(t_α)≠0; every nonzero x_α extends to an sl(2)-triple x_α,y_α,h_α with h_α=2t_α/κ(t_α,t_α), and h_{−α}=−h_α.
Proposition: integrality and root strings
For α∈Φ, dim L_α=1 and only ±α are scalar-multiple roots. For α,β∈Φ, the Cartan integer β(h_α) is integral and reflection sends β to β−β(h_α)α. Root strings β−rα,…,β+qα are unbroken with β(h_α)=r−q; if α+β is a root then [L_α,L_β]=L_{α+β}; the root spaces generate L.
Definition: Cartan integers
The integers β(h_α) attached to roots α,β are called Cartan integers.
Construction: rational and Euclidean root spaces
The Q-span E_Q of Φ has dimension dim H*, the Killing-induced form is rational and positive definite on E_Q, and E=R⊗_Q E_Q is a Euclidean space containing Φ.
Theorem: root-system axioms for the roots of a semisimple Lie algebra
In the Euclidean space E, Φ spans E and excludes 0; the only scalar multiples of α∈Φ are ±α; reflection σ_α(β)=β−2(β,α)/(α,α) α preserves Φ; and 2(β,α)/(α,α)∈Z. Thus Φ is a root system.