The Lie bracket of fundamental vector fields

[ubugnu]

Homework Statement

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:

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

Homework Equations

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

\sigma_{p}(X)=(p\,e^{tX})'(0)

The Attempt at a Solution

[\sigma(X),\sigma(Y)](f) = \sigma(X)[\sigma(Y)f]-X\leftrightarrow Y
= \sigma(X)[f(pe^{tY})'(0)]-X\leftrightarrow Y
= f(pe^{tX}e^{tY})'(0)-X\leftrightarrow Y
\sigma([X,Y])(f) = f(pe^{t[X,Y]})'(0)

 

[qinglong.1397]

Hope there is someone answering this problem soon :)