37: Tangent Vectors Space of General Linear Group of Finite-Dimensional Real Vectors Space at Identity Is 'Vectors Spaces - Linear Morphisms' Isomorphic to General Linear Lie Algebra

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description/proof of that tangent vectors space of general linear group of finite-dimensional real vectors space at identity is 'vectors spaces - linear morphisms' isomorphic to general linear Lie algebra

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Topics

About: general linear group of vectors space
About: general linear 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

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Starting Context

• The reader knows a definition of general linear group of vectors space.
• The reader knows a definition of general linear Lie algebra.
• The reader knows a definition of %category name% isomorphism.
• The reader knows a definition of tangent vectors space at point.
• The reader knows a definition of derivative of real-1-parameter family of vectors in finite-dimensional real vectors space.
• The reader admits the proposition that any bijective linear morphism is a 'vectors spaces - linear morphisms' isomorphism.

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Target Context

• The reader will have a description and a proof of the proposition that the tangent vectors space of the general linear group of any finite-dimensional real vectors space at the identity is 'vectors spaces - linear morphisms' isomorphic to the general linear Lie algebra, and will understand what the isomorphism is like.

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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:
$V$: $\in \{\text{ the finite-dimensional real vectors spaces }\}$
$GL(V)$: $=\text{ the general linear group of }V$ with the canonical $C^{\mathrm{\infty }}$ manifold structure
$\mathfrak{gl}(V)$: $=\text{ the general linear Lie algebra of }V$
$f$: $T_{I}GL(V)\to \mathfrak{gl}(V)$ such that $f(dc(t)/dt{|}_{t=0})(v)=d(c(t)(v))/dt{|}_{t=0}$, where $c:(-ϵ,ϵ)\to GL(V)$ is any $C^{\mathrm{\infty }}$ curve that realizes a tangent vector in $T_{I}GL(V)$, $v$ is any vector in $V$, and $d(c(t)(v))/dt{|}_{t=0}$ is the derivative of real-1-parameter family of vectors in finite-dimensional real vectors space
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Statements:
$f\in \{\text{ the 'vectors spaces - linear morphisms' isomorphisms }\}$
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2: Natural Language Description

For any finite-dimensional real vectors space, $V$, the general linear group, $GL(V)$, with the canonical $C^{\mathrm{\infty }}$ manifold structure, and the general linear Lie algebra, $\mathfrak{gl}(V)$, there is the 'vectors spaces - linear morphisms' isomorphism, $f:T_{I}GL(V)\to \mathfrak{gl}(V)$, such that $f(dc(t)/dt{|}_{t=0})(v)=d(c(t)(v))/dt{|}_{t=0}$, where $c:(-ϵ,ϵ)\to GL(V)$ is any $C^{\mathrm{\infty }}$ curve that realizes a tangent vector in $T_{I}GL(V)$, $v\in V$ is any vector, and $d(c(t)(v))/dt{|}_{t=0}$ is the derivative of real-1-parameter family of vectors in finite-dimensional real vectors space.

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3: Proof

$GL(V)$ as the canonical $C^{\mathrm{\infty }}$ manifold has the chart that maps onto the set of the non-singular matrices with respect to any fixed basis of $V$. Let us use the chart hereafter.

$\mathfrak{gl}(V)$ can be represented as the matrices with respect to the same fixed basis of $V$. Let us use the representation hereafter.

For any $l\in \mathfrak{gl}(V)$, let $L$ be the representation matrix. Let $c:(-ϵ,ϵ)\to GL(V)$ with $ϵ$ small enough be the $C^{\mathrm{\infty }}$ curve on $GL(V)$ whose chart components are $C(t):=I+tL$, which is indeed on $GL(V)$, because $C(t)$ is a parameterized matrix, $\left(\begin{array}{c}{C}_k^j(t)\end{array}\right)$, but $det\left(\begin{array}{c}{C}_k^j(t)\end{array}\right)$ is not $0$ there (it is $1$ at $t=0$ and is continuous with respect to $t$), so, $C(t)$ is invertible. But $S:=\{C(t):=I+tL|\mathrm{\forall }l\in \mathfrak{gl}(V)\}$ uniquely identifies the tangent vectors in $T_{I}GL(V)$, because each tangent vector is realized by an $l$ and for each $l_1\ne l_2$, the realized tangent vectors are different.

Let $f$ map each tangent vector to $l$ via $S$.

$f$ is surjective, because for any $l\in \mathfrak{gl}(V)$, there is the tangent vector realized by $C(t):=I+tL$, and is injective, because for any distinct 2 tangent vectors, the representing $l$ s are different.

$f$ is linear, because for any tangent vectors, $v_1,v_2$, such that $f(v_j)=l_j$, $r_1{v}_1+r_2{v}_2$ is realized by $C(t):=I+t(r_1{L}_1+r_2{L}_2)$, and $f(r_1{v}_1+r_2{v}_2)=r_1{l}_1+r_2{l}_2=r_{1}f(v_1)+r_{2}f(v_2)$.

So, $f$ is a 'vectors spaces - linear morphisms' isomorphism, by the proposition that any bijective linear morphism is a 'vectors spaces - linear morphisms' isomorphism.

Now, $f(dc(t)/dt{|}_{t=0})(v)=lv$, which is represented by $L{v}^{\prime }$ where $v^{\prime }$ is the representation of $v$ by the basis of $V$, $=\left(\begin{array}{c}d{C}_k^j(t)/dt{|}_{t=0}\end{array}\right)v^{\prime }=d/dt{|}_{t=0}(\left(\begin{array}{c}{C}_k^j(t)\end{array}\right)v^{\prime })$, which is the representation of $d(c(t)(v))/dt{|}_{t=0}$, which is the derivative of the real-1-parameter family of vectors in finite-dimensional real vectors space, which means that $f(dc(t)/dt{|}_{t=0})(v)=d(c(t)(v))/dt{|}_{t=0}$.

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References

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