definition of left-invariant vectors field over Lie group
Topics
About: group
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of Lie group.
- The reader knows a definition of \(C^\infty\) vectors field over \(C^\infty\) manifold with boundary.
- The reader knows a definition of map-related vectors fields pair for \(C^\infty\) map between \(C^\infty\) manifolds with boundary.
Target Context
- The reader will have a definition of left-invariant vectors field over Lie group.
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:
\( G\): \(\in \{\text{ the Lie groups }\}\)
\(*V\): \(\in \{\text{ the vectors fields over } G\}\)
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Conditions:
\(\forall g \in G ((V, V) \in \{\text{ the } l_g \text{ -related vectors fields pairs }\})\), where \(l_g: G \to G, g' \mapsto g g'\) is the left-translation by \(g\)
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2: Note
\(V\) is inevitably \(C^\infty\), by the proposition that any left-invariant vectors field over any Lie group is \(C^\infty\).
