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 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.