# The Lie bracket of fundamental vector fields

1. May 3, 2013

### ubugnu

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:

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

2. Relevant 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:

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

3. The attempt at a solution

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

2. Aug 23, 2013

### qinglong.1397

Hope there is someone answering this problem soon :)