The Lie bracket of fundamental vector fields

  1. 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. jcsd
  3. Hope there is someone answering this problem soon :)
     
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