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Divergence of left invariant vector field 
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#1
Aug211, 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? 


#2
Aug1711, 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?



#3
Aug1711, 01:27 PM

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PF Gold
P: 2,603

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. 


#4
Aug1811, 04:39 AM

P: 256

Divergence of left invariant vector field
Thanks fzero. I like your reasoning.



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