Lie Algebra
Theorem 1.
February 13, 2017
(Fourth Isomorphism Theorem) Let
Jakelic
I E L. Then A := {ideals of L containing I} ↔1−1correspondence
B := {ideals of L/I } .
Dene f1 : A → B such that f1 : J 7→ J/I where I ≤ J E L. Note this is
well dened because the third isomorphism theorem suggests J/I E L/I . Also, dene
f2 : B → A given by f2 : M 7→ {x ∈ L| x + I ∈ M } := J where M E L/I . We need
to show J E L. Containment is ne. J is a subspace. Take some x ∈ J , y ∈ L.
Then
Proof.
[x + I , y + I] = [x, y] + I ∈ M ⇒ [x, y] ∈ J.
| {z } | {z }
∈M
∈L/I
Thus, J E L. Show two way composition is the identity ⇒ bijective relation.
(1)