description/proof of that kernel of Lie algebra homomorphism is ideal of domain
Topics
About: Lie algebra
The table of contents of this article
Starting Context
- The reader knows a definition of kernel of linear map.
- The reader knows a definition of Lie algebra.
- The reader knows a definition of %structure kind name% homomorphism.
- The reader knows a definition of ideal of Lie algebra.
- The reader admits the proposition that the kernel of any linear map between any vectors spaces is a vectors subspace of the domain.
Target Context
- The reader will have a description and a proof of the proposition that the kernel of any Lie algebra homomorphism is an ideal of the domain.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(F\): \(\in \{\text{ the fields }\}\)
\(V_1\): \(\in \{\text{ the } F \text{ Lie algebras }\}\)
\(V_2\): \(\in \{\text{ the } F \text{ Lie algebras }\}\)
\(f\): \(: V_1 \to V_2\), \(\in \{\text{ the Lie algebra homomorphisms }\}\)
\(Ker (f)\): \(= \text{ the kernel of } f\)
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Statements:
\(Ker (f) \in \{\text{ the ideals of } V_1\}\)
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2: Proof
Whole Strategy: Step 1: see that \(Ker (f)\) is a vectors subspace of \(V_1\); Step 2: see that \(Ker (f)\) satisfies the conditions to be an ideal.
Step 1:
\(Ker (f)\) is a vectors subspace of \(V_1\), by the proposition that the kernel of any linear map between any vectors spaces is a vectors subspace of the domain.
Step 2:
Let \(v \in V_1\) and \(k \in Ker (f)\) be any.
\(f ([v, k]) = [f (v), f (k)]\), because \(f\) is a Lie algebra homomorphism, \(= [f (v), 0] = 0\), which means that \([v, k] \in Ker (f)\).
So, \(Ker (f)\) is an ideal of \(V_1\).
