Divergence of left invariant vector field

[paweld]

Let's assume that a compact Lie group and left invariant vector filed X are given.
I wonder why the divergence (with respect to Haar measure) of this field has to
be equall 0. I found such result in one paper but I don't know how to prove it.
Any suggestions?

 

[mathwonk]

searching the web i have found assertions that this holds for unimodular groups, but perhaps not in general. could there be another hypothesis you haven't mentioned?

 

[fzero]

Given a volume form $\omega$, then the divergence of a vector field $X$ is related to the Lie derivative as:

$(\mathrm{div}~X) \omega = \mathcal{L}_X \omega.$

In the case of a Lie group, there are presumably a number of ways to show that the RHS vanishes iff $X$ is an element of the Lie algebra. In particular, we can argue that it's natural from the point of view of the Lie algebra generating the isometries of the Lie group manifold.

 

[paweld]

Thanks fzero. I like your reasoning.