A description/proof of that point on connected Lie group can be expressed as finite product of exponential maps
Topics
About: Lie group
About: vectors field
The table of contents of this article
Starting Context
- The reader knows a definition of Lie group.
- The reader knows a definition of connected topological space.
- The reader knows a definition of Lie algebra.
- The reader knows a definition of exponential map on Lie group.
- The reader admits the proposition that any 2 points on any connected Lie group can be connected by a finite number of segments each of which is of an integral curve of a left-invariant vectors field.
- The reader admits the proposition that the integral curve of any left-invariant vectors field starting at any point on any Lie group is the integral curve of the vectors field starting at e, left-multiplied by the point.
Target Context
- The reader will have a description and a proof of the proposition that any point on any connected Lie group can be expressed as the product of some finite number of exponential maps.
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: Description
For any connected Lie Group, G, any point, \(p \in G\), can be expressed as \(p = exp (V_{0, e}) exp (V_{1, e}) . . . exp (V_{k, e})\).
2: Proof
p can be connected with e by a finite number of segments each of which is an integral curve of a left-invariant vectors field via \(p_0 := e \rightarrow p_1 \rightarrow . . . \rightarrow p_{k + 1} := p\). \(p_{i + 1} = p_i exp V_{i, e}\). \(p = p_{k + 1} = p_0 exp V_{0, e} exp V_{1, e} . . . exp V_{k, e}\).
