Is the exterior covariant derivative an anti-derivation?

From: Mark Adams (markjadams_at_lycos.com)
Date: 02/16/05 

Date: Wed, 16 Feb 2005 17:36:38 +0000 (UTC)

Hello,

I am trying to find out if the exterior covariant derivative is an
anti-derivation when applied to two types of Lie algebra valued forms.
The definition I am using of an anti-derivation is that for Lie algebra
valued forms a and b, the identity

D([a, b]) = [Da, b] + (-1)^k [a, Db]

is satisfied, where a is a k-form. The two cases I am interested in are:

(1) a and b take values in the same Lie algebra as the connection 1-form
A, so that

Da := da + [A, a]

and the Lie algebra valued form [a, b] is defined so that e.g. if a and
b are 1-forms we have

[a, b](u, v) := [a(u), b(v)] - [a(v), b(u)].

Thus the question is whether

D([a, b]) = d[a, b] + [A, [a, b]]

is equal to

[Da, b] + (-1)^k [a, Db] = ([da, b] + [[A, a], b]) + (-1)^k ([a, db] +
[a, [A, b]]).

(2) a and b take values in the vector space acted on by a representation
of the connection 1-form A, so that

Da := da + A ^ a

and the vector valued form [a, b] is defined as above, while e.g. for a
1-form a we have

(A ^ a)(u, v) := A(u)a(v) - A(v)a(u).

Thus the question is whether

D([a, b]) = d[a, b] + A ^ [a, b]

is equal to

[Da, b] + (-1)^k [a, Db] = ([da, b] + [A ^ a, b]) + (-1)^k ([a, db] +
[a, A ^ b]).

What would help here is some identities for expressions involving
vector-valued forms like d[a, b].
Thanks very much for any help or especially references.

Relevant Pages

  • exterior covariant derivative
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  • Re: exterior covariant derivative
    ... > Thus the requirement for an anti-derivation for such forms becomes ... > where a is a Lie algebra valued k-form. ... where A ^ a is computed using the matrix action of the Lie algebra on ...
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