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
Topics
About: general linear group of vectors space
About: general linear 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
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.
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.
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:
\(V\): \(\in \{\text{ the finite-dimensional real vectors spaces }\}\)
\(GL (V)\): \(= \text{ the general linear group of } V\) with the canonical \(C^\infty\) manifold structure
\(\mathfrak{gl} (V)\): \(= \text{ the general linear Lie algebra of } V\)
\(f\): \(T_IGL (V) \to \mathfrak{gl} (V)\) such that \(f (d c (t) / d t \vert_{t = 0}) (v) = d (c (t) (v)) / d t \vert_{t = 0}\), where \(c: (-\epsilon, \epsilon) \to GL (V)\) is any \(C^\infty\) curve that realizes a tangent vector in \(T_IGL (V)\), \(v\) is any vector in \(V\), and \(d (c (t) (v)) / d t \vert_{t = 0}\) 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 \(C^\infty\) manifold structure, and the general linear Lie algebra, \(\mathfrak{gl} (V)\), there is the 'vectors spaces - linear morphisms' isomorphism, \(f: T_IGL (V) \to \mathfrak{gl} (V)\), such that \(f (d c (t) / d t \vert_{t = 0}) (v) = d (c (t) (v)) / d t \vert_{t = 0}\), where \(c: (-\epsilon, \epsilon) \to GL (V)\) is any \(C^\infty\) curve that realizes a tangent vector in \(T_IGL (V)\), \(v \in V\) is any vector, and \(d (c (t) (v)) / d t \vert_{t = 0}\) is the derivative of real-1-parameter family of vectors in finite-dimensional real vectors space.
3: Proof
\(GL (V)\) as the canonical \(C^\infty\) 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 \(c: (-\epsilon, \epsilon) \to GL (V)\) with \(\epsilon\) small enough be the \(C^{\infty}\) curve on \(GL (V)\) whose chart components are \(C (t) := I + t L\), which is indeed on \(GL (V)\), because \(C (t)\) is a parameterized matrix, \(\begin{pmatrix} C^j_k (t) \end{pmatrix}\), but \(det \begin{pmatrix} C^j_k (t) \end{pmatrix}\) is not \(0\) there (it is \(1\) at \(t = 0\) and is continuous with respect to \(t\)), so, \(C (t)\) is invertible. But \(S := \{C (t) := I + t L \vert \forall l \in \mathfrak{gl} (V)\}\) uniquely identifies the tangent vectors in \(T_IGL (V)\), because each tangent vector is realized by an \(l\) and for each \(l_1 \neq l_2\), 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 \(C (t): = I + t L\), 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, \(f (d c (t) / d t \vert_{t = 0}) (v) = l v\), which is represented by \(L v'\) where \(v'\) is the representation of \(v\) by the basis of \(V\), \(= \begin{pmatrix} d C^j_k (t) / d t \vert_{t = 0} \end{pmatrix} v' = d / d t \vert_{t = 0} (\begin{pmatrix} C^j_k (t) \end{pmatrix} v')\), which is the representation of \(d (c (t) (v)) / d t \vert_{t = 0}\), which is the derivative of the real-1-parameter family of vectors in finite-dimensional real vectors space, which means that \(f (d c (t) / d t \vert_{t = 0}) (v) = d (c (t) (v)) / d t \vert_{t = 0}\).
