Chapter III. Lie Algebras
FC16 — Humphreys, Linear Algebraic Groups.
This chapter contains 5 definitions, 2 theorems, 6 propositions, 3 lemmas, 5 corollaries, 1 example, 1 construction and 1 other statements.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition: left-invariant derivations
A derivation of K[G] is left invariant when it commutes with left translations; these form L(G), the Lie algebra of G under commutator.
Theorem: Lie algebra as tangent space at identity
Evaluation at e gives a vector-space isomorphism L(G)≅T(G)_e, functorial for algebraic-group morphisms and compatible with Lie brackets.
Definition/construction: right convolution action of the tangent Lie algebra
A tangent vector x∈g acts on K[G] by right convolution f↦f*x; the commutator of convolution operators defines the Lie bracket on g.
Examples: Lie algebras of classical matrix groups
Lie(GL_n)=gl_n; Lie(SL_n)=sl_n; triangular, diagonal and unipotent triangular groups have the expected matrix Lie algebras, with analogous classical formulas.
Result: closed subgroups give Lie subalgebras
For a closed subgroup H≤G, h=T(H)_e embeds naturally as a Lie subalgebra of g.
Lemma: Lie algebra of a subgroup from its vanishing ideal
If I⊆K[G] vanishes on H, then h={x∈g | I*x⊆I}.
Definition: dual numbers
K[ε]=K[T]/(T²) is the algebra of dual numbers.
Construction: tangent group via dual numbers
For a group functor G, G(K[ε]) forms its tangent bundle; the kernel over G(K) identifies with the additive group of the Lie algebra.
Proposition: differentials of multiplication, inversion and commutator
At the identity, dμ(x,y)=x+y and di(x)=−x; the differential of the commutator morphism y↦yxy{-1}x{-1} is (1−Ad x).
Proposition: differential of right translation equals convolution
On a finite-dimensional right-translation-stable subspace of K[G], the differential of right translation is the right-convolution action of g.
Definition: adjoint representation
For x∈G, Ad x=d(Int x)_e∈GL(g).
Lemma: coordinate formulas for translations and convolution on GLn
For x∈GL_n and the coordinate functions T_ij, right translation, left translation and right convolution have the matrix-entry formulas stated in Lemma A.
Lemma: matrix formula for Ad on GLn
For x∈GL_n and y∈gl_n, Ad x(y)=xyx^{-1}.
Corollary: adjoint representation of GLn is algebraic
Ad:GL_n→GL(gl_n) is a morphism of algebraic groups.
Proposition: Ad is a rational representation
Ad:G→GL(g) is a morphism of algebraic groups; for matrix groups it is ordinary conjugation.
Definition: infinitesimal adjoint representation
ad:g→gl(g) sends x to the bracket operator y↦[x,y].
Theorem: differential of Ad is ad
d(Ad)_e=ad.
Corollary: normal subgroup gives Lie ideal
If H is a closed normal subgroup of G, then h is an ideal of g.
Corollary: Lie algebra of normalizer lies in Lie normalizer
For closed H≤G, Lie(N_G(H))⊆n_g(h)={x | [x,h]⊆h}; equality is later obtained in characteristic zero.
Proposition: Lie content of commutator subgroup closure
For closed A,B, the Lie algebra of the closure of (A,B) contains y−Ad x(y), x−Ad y(x), and [x,y] for x∈a,y∈b.
Corollary: derived Lie algebra lies in Lie of derived group closure
Lie(closure(G,G)) contains [g,g].
Proposition: Lie algebra of an element centralizer
Lie(C_G(x))⊆g^{Ad x}; equality holds for G=GL_n and later more generally in characteristic zero.
Proposition: differentiating dual and tensor representations
The dual action differentiates by (x·f)(v)=−f(x·v), while tensor products differentiate by Leibniz: x·(v⊗w)=xv⊗w+v⊗xw.
Corollary: Lie algebra of an algebra automorphism group consists of derivations
For a finite-dimensional K-algebra A, the Lie algebra of a closed subgroup of GL(A) acting by algebra automorphisms lies in Der(A).