Re: Is the exterior covariant derivative an anti-derivation?

From: Mark Adams (markjadams_at_lycos.com)
Date: 02/20/05 

Date: Sun, 20 Feb 2005 23:04:55 +0000 (UTC)

"Igor Khavkine" <igor.kh@gmail.com> wrote in message
news:pan.2005.02.17.05.03.52.172287@gmail.com...
> I think what you want to say here
> is that D is a *graded* derivation. And the identity you want to
satisfy
> is the graded Leibniz rule.

Yes, the term "antiderivation" as I'm using it means a graded derivation
of odd degree. For a and b taking values in a graded algebra (e.g. the
exterior algebra), a graded derivation satisfies the graded Leibniz rule

D(ab) = (Da)b + (-1)^ck a(Db)

where a is a k-form and the derivation is of degree c (e.g. in the
exterior algebra it takes k-forms to (k+c)-forms). Since the exterior
covariant derivative is of odd degree +1, if it satisfies the graded
Leibniz rule it would be an "antiderivation".

It's weird, most books I've found that treat the exterior covariant
derivative in detail seem to also introduce the idea of a derivation,
but none state whether D is a derivation or not...