Chapter VII. Chevalley Algebras and Groups
FC14 — Humphreys, Introduction to Lie Algebras and Representation Theory.
This chapter contains 14 definitions, 7 theorems, 6 propositions, 9 lemmas, 2 corollaries, 2 examples, 5 constructions and 1 convention.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Convention: Chevalley-theory setting
L is semisimple over algebraically closed F of characteristic zero, H a CSA, and Φ the root system. The chapter constructs integral forms and their reductions over arbitrary fields.
Proposition: rank-two root-string arithmetic
For independent roots α,β with α-string β−rα,…,β+qα: ⟨β,α⟩=r−q; at most two root lengths occur in the string; and if α+β is a root then the root-length/string relation in part (c) gives the integer needed for Chevalley structure constants.
Lemma: sl2 calculation along a root string
Choose x_α,x_{−α} with [x_α,x_{−α}]=h_α. For x_β∈L_β and α-string β−rα,…,β+qα, one has [x_{−α},[x_α,x_β]]=q(r+1)x_β.
Proposition: compatible choice of root vectors
Root vectors x_α can be chosen so [x_α,x_{−α}]=h_α and, when α,β,α+β are roots, structure constants c_{αβ} in [x_α,x_β]=c_{αβ}x_{α+β} satisfy c_{αβ}=−c_{−α,−β}; their squares/magnitudes are determined by root strings and root lengths.
Definition: Chevalley basis
A Chevalley basis is {x_α | α∈Φ}∪{h_i | 1≤i≤l}, where h_i=h_{α_i} for a base and the root vectors satisfy the compatibility conditions of the preceding proposition.
Theorem: Chevalley integral structure constants
For a Chevalley basis all structure constants are integers: [h_i,h_j]=0; [h_i,x_α]=⟨α,α_i⟩x_α; [x_α,x_{−α}]=h_α is an integral combination of h_i; and for independent α,β, [x_α,x_β]=0 when α+β is absent, otherwise ±(r+1)x_{α+β}.
Proposition: rescaling freedom of Chevalley root vectors
Replacing x_α by η(α)x_α preserves the Chevalley normalization precisely when η(α)η(−α)=1 and η(α)η(β)=±η(α+β) compatibly with the structure-constant signs. The absolute constants are root-system determined, while signs require a coherent choice.
Definition/construction: integral Chevalley form L(Z)
The Z-span L(Z) of a Chevalley basis is a free Z-lattice in L and a Lie algebra over Z. It is independent of the chosen base and, up to Z-isomorphism, of the Chevalley basis.
Definition/construction: Chevalley algebra over a field
For a field K of characteristic p, define L(K)=L(Z)⊗_Z K (equivalently reduce first to F_p and extend scalars). It inherits the Chevalley multiplication table modulo p and is called a Chevalley algebra.
Example: type A Chevalley algebra
For L=sl(l+1,F), the Chevalley algebra L(K) is sl(l+1,K). It may cease to be simple when char K divides l+1, acquiring scalar center.
Proposition: divided powers of ad xα preserve L(Z)
For α∈Φ and m≥0, (ad x_α)^m/m! maps L(Z) into L(Z); on root strings its coefficients are binomial integers.
Construction: integral inner automorphisms and reduction modulo p
Because the divided powers preserve L(Z), exp(ad x_α) has an integral matrix in a Chevalley basis. The subgroup generated by exp(ad c x_α), c∈Z, preserves L(Z), and reducing matrices modulo p acts by automorphisms of L(F_p).
Definition/construction: Chevalley group of adjoint type
Using exp(ad T x_α) over Z[T] and specializing T to c∈K, the subgroup G(K) generated by the resulting root elements is the Chevalley group of adjoint type attached to Φ. For finite K it is finite and, apart from specified small exceptions, simple.
Definition: polynomial binomial coefficient
For an element x in a commutative F-algebra and k≥0, the polynomial binomial coefficient binom(x,k)=x(x−1)⋯(x−k+1)/k! is defined and obeys the usual finite-difference identity.
Lemma: integer-valued polynomials in several variables
If f(T_1,…,T_l)∈F[T_1,…,T_l] takes integer values on Z^l, then it is a Z-linear combination of products ∏_i binom(T_i,b_i), with b_i bounded by the degree in T_i.
Lemma: divided-power commutation formula for sl2
For the standard sl2 generators x,y,h and nonnegative a,c, the source gives an explicit formula rewriting x^c/c! · y^a/a! as a finite sum of ordered divided powers y^{a−k}/(a−k)!, polynomial binomials in h, and x^{c−k}/(c−k)!, all with integral coefficients.
Corollary: binomial polynomials in h lie in the integral divided-power subring
For every b≥0, binom(h,b) lies in the subring of U(sl2) generated by all divided powers x^c/c! and y^a/a!.
Lemma: tensor products preserve divided-power-stable lattices
If subgroups A⊂V and B⊂W are invariant under x_α^t/t! for all roots α and t≥0, then A⊗B is likewise invariant.
Corollary: tensor powers of L(Z) are divided-power stable
Every tensor power L(Z)^{⊗n} is invariant under the adjoint divided powers (ad x_α)^t/t!.
Definition: closed set of roots
A subset Ψ⊆Φ is closed if α,β∈Ψ and α+β∈Φ imply α+β∈Ψ.
Lemma: integral PBW basis for a closed positive root set
If Ψ is closed and Ψ∩(−Ψ)=∅, then the subring of U(L) generated by x_α^t/t!, α∈Ψ, has as Z-basis the ordered products of these divided powers.
Definition: height of a divided-power/binomial monomial
A monomial in U(L) built from root divided powers and binomials in h_i has height equal to the sum of the divided-power exponents.
Lemma: commuting divided powers of root vectors
For roots α,β and k,m≥0, (x_αk/k!)(x_βm/m!) is a Z-linear combination of the reverse ordered product and other divided-power monomials of height <k+m.
Lemma: commuting root divided powers with Cartan polynomials
For f(T)∈F[T], roots α,β and k≥0, x_α^k f(h_β)=f(h_β−kα(h_β))x_α^k.
Construction: Kostant PBW basis elements
Fix orders on positive roots. For exponent tuples A,B,C define f_A as ordered negative-root divided powers, h_B as products of binom(h_i,b_i), and e_C as ordered positive-root divided powers. The products f_A h_B e_C form an F-basis of U(L).
Definition: Kostant integral form U(L)Z
U(L)_Z is the unital subring of U(L) generated by all x_α^t/t!, α∈Φ, t≥0.
Theorem: Kostant’s integral form theorem
The Z-lattice with basis {f_A h_B e_C} is exactly U(L)_Z. Thus Kostant’s integral form is a free Z-module with this triangular PBW basis.
Definition: integral forms of U(N−), U(H), U(N+)
The corresponding Z-forms U_Z^−, U_Z^0, U_Z^+ have bases f_A,h_B,e_C and satisfy U(L)_Z=U_Z^− U_Z^0 U_Z^+.
Definition: lattice in a finite-dimensional F-vector space
A lattice M⊂V is the Z-span of an F-basis; equivalently a finitely generated subgroup spanning V over F and of Z-rank at most dim_F V.
Lemma: integer-valued polynomial projection lemma
Given d∈Z^l and finite S⊂Z^l not containing d, there is an integer-valued polynomial f with f(d)=1 and f(S)=0.
Theorem: existence and weight decomposition of admissible lattices
For finite-dimensional L-module V: any subgroup stable under U(L)_Z is the direct sum of its intersections with weight spaces; and V contains a U(L)_Z-stable lattice.
Definition: admissible lattice
A lattice in a finite-dimensional L-module invariant under U(L)_Z is called admissible. For irreducible V(λ), U(L)_Z v^+ is the minimal admissible lattice containing a chosen maximal vector.
Definition: weight lattice A(V) of a faithful module
For a faithful finite-dimensional L-module V, A(V) is the Z-span of its weights; it lies between the root lattice and full weight lattice.
Definition/construction: stabilizer lattice LV of an admissible lattice
For admissible M⊂V, L_V={x∈L | xM⊆M}; put H_V=H∩L_V. One has L(Z)⊆L_V and H(Z)⊆H_V⊆H determined contravariantly by the lattice A(V).
Lemma: adjoint divided-power formula on U(L)
For u∈U(L), x∈L and n≥0, (ad x)n(u)/n!=Σ_{i=0}n (−1)^i x^{n−i}/(n−i)! · u · x^i/i!.
Proposition: structure of the stabilizer lattice
L_V is an admissible lattice in the adjoint L-module and L_V=H_V⊕⊕_{α∈Φ}Zx_α. Hence L_V depends only on V, equivalently on A(V), and not on the chosen admissible lattice M.
Example: two stabilizer lattices for sl2
For the natural two-dimensional sl2-module, L_V=L(Z); for the adjoint module, the Cartan lattice is enlarged to Z(h/2), illustrating the two possible A_1 weight lattices.
Construction: dual of an admissible lattice
For admissible M⊂V, M={f∈V | f(M)⊆Z} is an admissible lattice in the contragredient module; inclusions reverse under duality.
Proposition: minimal and maximal admissible lattices in V(λ)
Fix a maximal vector v^+ in irreducible V(λ). Every admissible lattice meeting V_λ in Zv^+ contains the minimal lattice M_min=U(L)_Z v^+ and is contained in a canonical maximal lattice obtained by dualizing the corresponding minimal lattice in V(λ)^*.
Construction: reduction of modules and Lie algebras to an arbitrary field
For a field K of characteristic p and admissible M⊂V, set V(K)=M⊗_Z K and L_V(K)=L_V⊗_Z K. The inclusion L(Z)⊆L_V induces L(K)→L_V(K)⊆gl(V(K)); the map can acquire a central kernel in small characteristic.
Construction: root elements on V(K)
Because M is divided-power stable, each root α yields nilpotent endomorphisms x_{α,t} on V(K). The polynomial root element x_α(c)=Σ_t c^t x_{α,t} lies in SL(V(K)).
Definition: Chevalley group of type A(V)
G_V(K) is the subgroup of SL(V(K)) generated by all root elements x_α(c), α∈Φ,c∈K. It is of adjoint type if A(V) is the root lattice and universal type if A(V) is the full weight lattice.
Theorem/recorded result: dependence of Chevalley groups and Lie algebras only on the weight lattice
The source records that, up to isomorphism, G_V(K) and L_V(K) depend only on A(V), not on V or the admissible lattice. Inclusions of weight lattices induce canonical homomorphisms, with universal type covering all types and all types covering adjoint type.
Theorem/recorded result: highest-weight structure after reduction
For V(λ), using the minimal admissible lattice gives V(K) a unique maximal G_V(K)-submodule and unique irreducible highest-weight quotient; using the maximal admissible lattice gives a unique irreducible highest-weight submodule.
Theorem/recorded result: restricted irreducible modules in characteristic p
If 0≤λ(h_i)<p for all simple coroots, the corresponding reduced modules give the p^l inequivalent irreducible restricted modules for L(K) (or L_V(K)), exhausting their isomorphism classes.
Theorem/recorded result: composition factors independent of admissible lattice
The composition factors of V(K), as a G_V(K)-module or L_V(K)-module, are independent of the admissible lattice used for reduction.