935: Range of Lie Algebra Homomorphism Is Lie Subalgebra of Codomain

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description/proof of that range of Lie algebra homomorphism is Lie subalgebra 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: Proof

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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.

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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.

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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:
$F$: $\in \{\text{ the fields }\}$
$V_1$: $\in \{\text{ the }F\text{ Lie algebras }\}$
$V_2$: $\in \{\text{ the }F\text{ Lie algebras }\}$
$f$: $:V_1\to V_2$, $\in \{\text{ the Lie algebra homomorphisms }\}$
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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 $f(v),f(v^{\prime })\in f(V_1)$, $[f(v),f(v^{\prime })]=f([v,v^{\prime }])\in f(V_1)$.

Let $f(v),f(v^{\prime }),f(v^{″})\in f(V_1)$ be any. Let $r,r^{\prime }\in F$ be any.

1) $[rf(v)+r^{\prime }f(v^{\prime }),f(v^{″})]=r[f(v),f(v^{″})]+r^{\prime }[f(v^{\prime }),f(v^{″})]$ $\wedge$ $[f(v^{″}),rf(v)+r^{\prime }f(v^{\prime })]=r[f(v^{″}),f(v)]+r^{\prime }[f(v^{″}),f(v^{\prime })]$: it holds on ambient $V_2$.

2) $[f(v^{\prime }),f(v)]=-[f(v),f(v^{\prime })]$: it holds on ambient $V_2$.

3) $\sum _{cyclic}[f(v),[f(v^{\prime }),f(v^{″})]]=0$: it holds on ambient $V_2$.

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References

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