description/proof of that range under Lie algebra homomorphism is Lie sub-algebra of codomain
Topics
About: Lie algebra
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Proof
- 4: Note
Starting Context
- The reader knows a definition of Lie algebra.
Target Context
- The reader will have a description and a proof of the proposition that the range under any Lie algebra homomorphism is a Lie sub-algebra 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:
\(F\): \(\in \{\text{ the fields }\}\)
\(V_1\): \(\in \{\text{ the Lie algebras over } F\}\)
\(V_2\): \(\in \{\text{ the Lie algebras over } F\}\)
\(f\): \(: V_1 \to V_2\), \(\in \{\text{ the Lie algebra homomorphisms }\}\)
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Statements:
\(f (V_1) \in \{\text{ the Lie sub-algebras of } V_2\}\)
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2: Natural Language Description
For any field, \(F\), any Lie algebras over \(F\), \(V_1, V_2\), and any Lie algebra homomorphism, \(f: V_1 \to V_2\), \(f (V_1)\) is a Lie sub-algebra of \(V_2\).
3: Proof
Is \(f (V_1)\) an \(F\) vectors space? Is \(f (V_1)\) closed under the vectors space operations? If so, \(f (V_1)\) will be an \(F\) vectors space, because the properties of vectors space will hold there as they hold in the ambient \(V_2\). For any \(f (v_1), f (v_2) \in f (V_1)\) and any \(r_1, r_2 \in F\), \(r_1 f (v_1) + r_2 f (v_2) = f (r_1 v_1 + r_2 v_2) \in f (V_1)\). So, yes.
Is \(f (V_1)\) closed under the bracket? If so, \(f (V_1)\) will be a Lie algebra, because the properties of Lie algebra will hold there as they hold in the ambient \(V_2\). For any \(f (v_1), f (v_2) \in f (V_1)\), \([f (v_1), f (v_2)] = f ([v_1, v_2]) \in f (V_1)\). So, yes.
4: Note
Although this article has confirmed the proposition explicitly, in the 1st place, any homomorphism should be defined to be such that the range is the subspace of the codomain.
