The Lie bracket of fundamental vector fields


by ubugnu
Tags: differential geom, lie algebras, vector fields
ubugnu
ubugnu is offline
#1
May3-13, 08:44 AM
P: 3
1. The problem statement, all variables and given/known data

The Lie bracket of the fundamental vector fields of two Lie algebra elements is the fundamental vector field of the Lie bracket of the two elements:

[itex][\sigma(X),\sigma(Y)]=\sigma([X,Y])[/itex]

2. Relevant equations

Let [itex]\mathcal{G}[/itex] a Lie algebra, the fundamental vector field of an element [itex]X\in\mathcal{G}[/itex] is defined at a point [itex]p\in M[/itex] of a manifold [itex]M[/itex] as:

[itex]\sigma_{p}(X)=(p\,e^{tX})'(0)[/itex]

3. The attempt at a solution

[itex][\sigma(X),\sigma(Y)](f) = \sigma(X)[\sigma(Y)f]-X\leftrightarrow Y[/itex]
[itex] = \sigma(X)[f(pe^{tY})'(0)]-X\leftrightarrow Y[/itex]
[itex] = f(pe^{tX}e^{tY})'(0)-X\leftrightarrow Y[/itex]
[itex]\sigma([X,Y])(f) = f(pe^{t[X,Y]})'(0)[/itex]
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qinglong.1397
qinglong.1397 is offline
#2
Aug23-13, 04:11 PM
P: 107
Hope there is someone answering this problem soon :)


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