635: Range Under Lie Algebra Homomorphism Is Lie Sub-Algebra of Codomain

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description/proof of that range under Lie algebra homomorphism is Lie sub-algebra of codomain

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Topics

About: Lie algebra

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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

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Starting Context

• The reader knows a definition of Lie algebra.

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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.

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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.

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Main Body

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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 $V_2$.

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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.

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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.

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References

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