Chapter V. Existence Theorem
FC14 — Humphreys, Introduction to Lie Algebras and Representation Theory.
This chapter contains 9 definitions, 8 theorems, 4 propositions, 4 lemmas, 6 corollaries, 1 example, 7 constructions and 3 conventions.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Convention: field and dimension conventions for enveloping algebras
In §17 the field F is arbitrary; Lie algebras may be infinite-dimensional where explicitly allowed.
Definition/construction: tensor algebra
For a vector space V, T(V)=⊕_{m≥0}T^mV with T^0V=F, T^1V=V and TmV=V{⊗m}; concatenation of tensors makes T(V) an associative graded F-algebra with 1, universal for linear maps V→A into associative unital F-algebras.
Definition/construction: symmetric algebra
S(V)=T(V)/I where I is generated by x⊗y−y⊗x. It is the universal commutative associative unital algebra receiving V, is graded, and for a basis of V identifies canonically with a polynomial algebra.
Definition/result: homogeneous symmetric tensors
In characteristic zero, the symmetric group acts on T^mV by permuting tensor factors. Fixed tensors are homogeneous symmetric tensors of order m; averaging over permutations gives a subspace canonically isomorphic to S^mV and hence a graded vector-space copy of S(V) inside T(V).
Definition: universal enveloping algebra
A universal enveloping algebra of a Lie algebra L is an associative unital algebra U with a linear map i:L→U satisfying i([x,y])=i(x)i(y)−i(y)i(x), universal among such pairs. It is unique up to unique isomorphism.
Construction: U(L) as a quotient of T(L)
Let J be the two-sided ideal of T(L) generated by x⊗y−y⊗x−[x,y]. Then U(L)=T(L)/J with the induced map i:L→U(L) satisfies the universal property.
Example: enveloping algebra of an abelian Lie algebra
If L is abelian, U(L) is canonically the symmetric algebra S(L).
Construction: PBW filtration and associated graded algebra
Filter T(L) by tensor degree T_m and U(L) by U_m=π(T_m). The quotients G^m=U_m/U_{m−1} form the associated graded algebra G; the canonical T(L)→G kills the commutativity ideal and induces a surjective graded algebra map w:S(L)→G.
Theorem: Poincaré–Birkhoff–Witt theorem
The canonical graded algebra homomorphism w:S(L)→gr U(L) is an isomorphism.
Corollary: degree-m complements in the PBW filtration
If W⊆T^mL maps isomorphically to S^mL, then its image in U_m is a vector-space complement of U_{m−1}.
Corollary: canonical embedding of L into U(L)
The canonical map i:L→U(L) is injective, so L may be identified with its image in U(L).
Corollary: PBW basis
For any ordered basis (x_i) of L, 1 together with ordered monomials x_{i_1}⋯x_{i_m}, i_1≤⋯≤i_m, form a basis of U(L). Such a basis is called a PBW basis.
Corollary: enveloping algebra over a subalgebra
If H≤L and an ordered basis of H is extended to one of L, the induced map U(H)→U(L) is injective and U(L) is a free U(H)-module on the ordered monomials in the complementary basis elements.
Corollary: symmetric tensors as PBW filtration complements
In characteristic zero, the canonical map from homogeneous symmetric tensors SmL⊂TmL to U_m is an isomorphism onto a complement of U_{m−1}.
Lemma: PBW proof: recursive action on the symmetric algebra
For an ordered basis of L, the source constructs compatible linear maps f_m:L⊗S_m→S satisfying the order, filtration, and commutator recursion needed to define the PBW representation.
Lemma: PBW proof: representation on S(L)
The recursive maps assemble to a representation ρ:L→gl(S(L)) whose leading term is multiplication by the corresponding degree-one variable.
Lemma: PBW proof: highest homogeneous term of a relation
If t lies in the kernel J of T(L)→U(L), then its highest homogeneous component lies in the commutativity ideal I=Ker(T(L)→S(L)).
Theorem: PBW kernel comparison
The preceding representation proves PBW by showing that a homogeneous t∈T^mL whose image lies in U_{m−1} must lie in I; equivalently S(L)≅gr U(L).
Definition/construction: free Lie algebra
A Lie algebra L generated by X is free on X if every map X→M into a Lie algebra extends uniquely to a Lie homomorphism L→M. It exists as the Lie subalgebra generated by X inside the tensor algebra on the vector space with basis X.
Construction: modules over a free Lie algebra
To make V an L-module when L is free on X, it suffices to assign arbitrary endomorphisms of V to the generators X; the assignment extends uniquely to a representation.
Definition: Lie algebra by generators and relations
If L is free on X and R is the ideal generated by relations f_j, then L/R is called the Lie algebra with generators given by the images of X and relations f_j=0.
Convention: setting for Serre presentation
In §18, L is semisimple over an algebraically closed field of characteristic zero; infinite-dimensional intermediate Lie algebras are allowed.
Proposition: Chevalley–Serre relations satisfied by standard generators
For standard generators x_i,y_i,h_i associated with a base of Φ, L is generated by them and satisfies [h_i,h_j]=0; [x_i,y_j]=δ_{ij}h_i; [h_i,x_j]=c_{ji}x_j, [h_i,y_j]=−c_{ji}y_j; and the Serre nilpotence relations (ad x_i)^{1−c_{ji}}x_j=0 and (ad y_i)^{1−c_{ji}}y_j=0 for i≠j.
Construction: Lie algebra L0 defined by relations S1–S3
For a root system with Cartan matrix (c_{ij}), start from the free Lie algebra on generators X_i,Y_i,H_i and quotient only by the relations [h_i,h_j]=0, [x_i,y_j]=δ_{ij}h_i, [h_i,x_j]=c_{ji}x_j, [h_i,y_j]=−c_{ji}y_j. The resulting algebra is denoted L_0.
Construction: model representation of L0 on a tensor algebra
On the tensor algebra generated by v_1,…,v_l, the source defines explicit endomorphisms for h_i,y_i,x_i and obtains a representation of the free Lie algebra.
Proposition: the model representation factors through L0
The explicit representation annihilates the defining relations S1–S3, hence factors through L_0.
Theorem: triangular decomposition of L0
In L_0 the h_i form a basis of an l-dimensional abelian subalgebra H, and L_0=Y⊕H⊕X, where X and Y are the subalgebras generated by the x_i and y_i. Nonzero H-weights are integral combinations of simple roots with coefficients all positive or all negative.
Lemma: Serre generators are killed by opposite simple raising operators
For y_{ij}=(ad y_i)^{1−c_{ji}}y_j in L_0, ad x_k(y_{ij})=0 for every k; the analogous statement holds after exchanging x and y.
Definition/construction: locally nilpotent endomorphism and exponential
An endomorphism is locally nilpotent if every vector is killed by some power. Its exponential is defined by patching the finite exponential series on finite-dimensional invariant subspaces; this yields an automorphism.
Theorem: Serre’s Theorem
For a root system Φ with base Δ, the Lie algebra generated by x_i,y_i,h_i subject to S1–S3 and the Serre relations is finite-dimensional semisimple; H=span(h_i) is a Cartan subalgebra and its root system is Φ.
Corollary: dimension from the root system in Serre’s construction
The Lie algebra in Serre’s theorem has root decomposition H⊕⊕_{α∈Φ}L_α with dim H=rank Φ and dim L_α=1; hence dim L=rank Φ+|Φ|.
Theorem: existence of semisimple Lie algebras with prescribed root system
For every root system Φ there exists a semisimple Lie algebra having Φ as its root system.
Theorem: existence-and-uniqueness isomorphism theorem from Serre presentation
If semisimple L,L′ have isomorphic root systems and chosen compatible Cartan subalgebras/bases and nonzero simple-root vectors, there is a unique Lie-algebra isomorphism extending the Cartan isomorphism and the chosen root-vector identifications.
Convention: classification setting for simple algebras
In §19, F is algebraically closed of characteristic zero. Up to isomorphism the simple Lie algebras correspond to A_l,B_l,C_l,D_l and the exceptional types E_6,E_7,E_8,F_4,G_2, with standard rank restrictions.
Definition: reductive Lie algebra
A nonzero Lie algebra L is reductive if Rad L=Z(L). Abelian and semisimple algebras are the two extreme cases.
Proposition: structure and irreducible-action criterion for reductive Lie algebras
If L is reductive, L=[L,L]⊕Z(L), with [L,L] semisimple or zero. If nonzero L⊆gl(V) acts irreducibly, then L is reductive with dim Z(L)≤1; if moreover L⊆sl(V), then L is semisimple.
Proposition: classical Lie algebras are the simple types A–D
The classical algebras sl(l+1,F), o(2l+1,F), sp(2l,F), and o(2l,F) act irreducibly in their natural modules, are semisimple by the preceding criterion, and their diagonal Cartan subalgebras/root spaces identify them as the simple types A_l,B_l,C_l,D_l in the stated ranges.
Construction: 14-dimensional matrix model of G2
Inside o(7,F), the source explicitly spans a 14-dimensional Lie algebra L by a two-dimensional diagonal subalgebra H and twelve displayed long/short root vectors, with bracket relations closing L.
Theorem/example: matrix model is the simple Lie algebra G2
The constructed 14-dimensional L acts irreducibly on F^7, lies in sl(7,F), and is therefore semisimple; its rank-two root data force the irreducible root system G_2, so L is the simple algebra of type G_2.
Definition/construction: Cayley/octonion algebra model
The source defines an eight-dimensional nonassociative Cayley (octonion) algebra C as 2×2 block matrices built from scalars and F^3, with multiplication using dot and cross products; it records the identity and a seven-dimensional trace-zero commutator subspace C_0.
Construction: derivation algebra of the Cayley algebra
Der(C) acts faithfully on the seven-dimensional trace-zero subspace C_0 and trivially on scalars. The source constructs an embedded sl(3,F) of derivations and a two-dimensional diagonal subalgebra H that is self-centralizing.
Theorem/identification: G2 as the derivation algebra of the octonions
The source identifies the derivation algebra Der(C), via its faithful action on C_0, with the previously constructed 14-dimensional simple Lie algebra of type G_2; the remaining explicit derivations are deferred to exercises.