description/proof of that range of Lie algebra homomorphism is Lie subalgebra of codomain
Topics
About: Lie algebra
The table of contents of this article
Starting Context
- The reader knows a definition of Lie algebra.
- The reader knows a definition of %structure kind name% homomorphism.
- The reader admits the proposition that the range of any linear map between any vectors spaces is a sub-'vectors space' of the codomain.
Target Context
- The reader will have a description and a proof of the proposition that the range of any Lie algebra homomorphism is a Lie subalgebra of the codomain.
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:
\(F\): \(\in \{\text{ the fields }\}\)
\(V_1\): \(\in \{\text{ the } F \text{ Lie algebras }\}\)
\(V_2\): \(\in \{\text{ the } F \text{ Lie algebras }\}\)
\(f\): \(: V_1 \to V_2\), \(\in \{\text{ the Lie algebra homomorphisms }\}\)
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Statements:
\(f (V_1) \in \{\text{ the Lie subalgebras of } V_2\}\)
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2: Proof
Whole Strategy: Step 1: see that \(f (V_1)\) is an \(F\) vectors space; Step 2: see that \(f (V_1)\) satisfies the other requirements to be a Lie algebra.
Step 1:
\(f (V_1)\) is an \(F\) vectors space, by the proposition that the range of any linear map between any vectors spaces is a sub-'vectors space' of the codomain.
Step 2:
For any \(f (v), f (v') \in f (V_1)\), \([f (v), f (v')] = f ([v, v']) \in f (V_1)\).
Let \(f (v), f (v'), f (v'') \in f (V_1)\) be any. Let \(r, r' \in F\) be any.
1) \([r f (v) + r' f (v'), f (v'')] = r [f (v), f (v'')] + r' [f (v'), f (v'')]\) \(\land\) \([f (v''), r f (v) + r' f (v')] = r [f (v''), f (v)] + r' [f (v''), f (v')]\): it holds on ambient \(V_2\).
2) \([f (v'), f (v)] = - [f (v), f (v')]\): it holds on ambient \(V_2\).
3) \(\sum_{cyclic} [f (v), [f (v'), f (v'')]] = 0\): it holds on ambient \(V_2\).
