From: baez@galaxy.ucr.edu (John Baez)
Subject: Re: Leibniz for adjoint exterior derivative?
Date: 18 Aug 2000 15:15:23 GMT
Newsgroups: sci.physics.research
Summary: [missing]

In article <3999E753.28B56B99@uiuc.edu>, Eric Forgy  <forgy@uiuc.edu> wrote:

>The Leibniz rule I get is a bit strange. It is something like:
>
>#del#(A^B) = (#del#A)^B + (-1)^p(A^#del#B + [A,B])
>
>where A is a p-vector, B is a q-vector, # is a map from p-vectors to
>p-forms and [A,B] is something like a Lie bracket (I remember reading
>about Schouten-Nijenhuis bracket, don't remember exactly what it was, but
>it might be pertinent.) If A and B are 1-vectors, i.e. just regular
>vectors, then [A,B] IS the Lie bracket.

In case people are wondering, the Schouten-Nijenhuis bracket is the
answer to the question "if p-forms are so great because you can define
the exterior derivative of them, why isn't there something comparably
great about the dual thing: p-vector fields?"    

(Recall that a p-vector field is a section of the pth antisymmetrized
tensor power of the *tangent* bundle, just as a p-form is a section of
the pth antisymmetrized tensor power of the *cotangent* bundle.)

Given a p-vector field A and a q-vector field B, the Schouten-Nijenhuis
bracket [A,B] is a (p+q-1)-vector field.  That "-1" is a bit odd at first
sight: we say the bracket makes the multivector fields into a "graded 
Lie algebra of degree -1".  

I'm not familiar with the identity you write down, but maybe I should
be!  Since I'm too lazy to work it out, I'll start by asking: 

1) is that [A,B] thing of yours a (p+q-1)-vector field?

2) does it equal the usual Lie bracket when p = q = 1?  (Yes, you say.)

3) does it equal the derivative of the function B in the direction
of the vector field A when p = 1, q = 0?

4) is there some nice rule saying that [A,B] is +-[B,A]?

If so, we're almost done showing it's the Schouten-Nijenhuis bracket.
The only other thing to check would be a graded version of the Jacobi 
identity.  

Your question was very interesting to me all along, by the way - the
only reason I didn't say anything about it was that I didn't know the
answer!   There's got to be *some* nice answer; on that you can depend.