# Divergence of left invariant vector field

by paweld
Tags: divergence, field, invariant, vector
 P: 256 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?
 HW Helper Sci Advisor P: 9,371 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?
 PF Patron HW Helper Sci Advisor P: 2,595 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.
P: 256

## Divergence of left invariant vector field

Thanks fzero. I like your reasoning.

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