Divergence of left invariant vector field

by paweld
Tags: divergence, field, invariant, vector
paweld is offline
Aug2-11, 04:39 AM
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?
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mathwonk is offline
Aug17-11, 12:34 PM
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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 is offline
Aug17-11, 01:27 PM
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Given a volume form [itex]\omega[/itex], then the divergence of a vector field [itex]X[/itex] is related to the Lie derivative as:

[itex](\mathrm{div}~X) \omega = \mathcal{L}_X \omega.[/itex]

In the case of a Lie group, there are presumably a number of ways to show that the RHS vanishes iff [itex]X[/itex] 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 is offline
Aug18-11, 04:39 AM
P: 256

Divergence of left invariant vector field

Thanks fzero. I like your reasoning.

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