John Stillwell - Naive Lie Theory.pdf - Index of

6.2 Ideals and homomorphisms 121
Proof. Since ϕ preserves sums and scalar multiples, h is a subspace:
X 1 ,X 2 ∈ h ⇒ ϕ(X 1 )=0,ϕ(X 2 )=0
⇒ ϕ(X 1 + X 2 )=0 because ϕ preserves sums
⇒ X 1 + X 2 ∈ h,
X ∈ h ⇒ ϕ(X)=0
⇒ cϕ(X)=0
⇒ ϕ(cX)=0 because ϕ preserves scalar multiples
⇒ cX ∈ h.
Also, h is closed under Lie brackets with members of g because
X ∈ h ⇒ ϕ(X)=0
⇒ ϕ([X,Y]) = [ϕ(X),ϕ(Y )] = [0,ϕ(Y )] = 0
for any Y ∈ g because ϕ preserves Lie brackets
⇒ [X,Y ] ∈ h for any Y ∈ g.
Thus h is an ideal, as claimed.
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It follows from this theorem that a Lie algebra is not simple if it admits
a nontrivial homomorphism. This points to the existence of non-simple Lie
algebras, which we should look at first, if only to know what to avoid when
we search for simple Lie algebras.
Exercises
There is a sense in which any homomorphism of a Lie group G “induces” a homomorphism
of the Lie algebra T 1 (G). We study this relationship in some depth in
Chapter 9. Here we explore the special case of the det homomorphism, assuming
also that G is a group for which exp maps T 1 (G) onto G.
6.2.1 If we map each X ∈ T 1 (G) to Tr(X), where does the corresponding member
e X of G go?
6.2.2 If we map each e X ∈ G to det(e X ), where does the corresponding X ∈ T 1 (G)
go?
6.2.3 In particular, why is there a well-defined image of X when e X = e X′ ?