The Lie bracket of fundamental vector fieldsby ubugnu Tags: differential geom, lie algebras, vector fields 

#1
May313, 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] 



#2
Aug2313, 04:11 PM

P: 107

Hope there is someone answering this problem soon :)



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