- #1
center o bass
- 560
- 2
In the formulation of connections on principal bundles, one derives an
expression for the covariant exterior derivative of lie-algebra valued forms which is given by
$$D\alpha = d \alpha + \rho(\omega) \wedge \alpha,$$
where ##\rho: \mathfrak g \to \mathfrak{gl}(\mathfrak g)## is a representation on the Lie algebra. Now, one often encounters the following formula for the curvature of the connection ##\omega##:
$$\Omega = D\omega = d\omega + \frac{1}2 [\omega, \omega].$$
However, if we use the representation ##\text{ad}:\mathfrak g \to \mathfrak{gl}(\mathfrak g)##, then the covariant exterior derivative of ##\omega## gives
$$D\alpha = d \alpha + \rho(\omega) \wedge \alpha = d \alpha + [\xi_k, \xi_l] \omega^k \wedge \omega^l = d\alpha + [\omega, \omega]$$.
But where have the factor ##1/2## gone?
I suspect my error might lie in one of the following:
(1): my definition of ##[\omega, \omega]## as ##[\xi_k, \xi_l] \omega^k \wedge \omega^l##.
(2): or, that what is meant by the expression ##\rho(\omega) \wedge \omega## is perhaps not ##[\xi_k, \xi_l] \omega^k \wedge \omega^l##.
But which one is it? And why?
expression for the covariant exterior derivative of lie-algebra valued forms which is given by
$$D\alpha = d \alpha + \rho(\omega) \wedge \alpha,$$
where ##\rho: \mathfrak g \to \mathfrak{gl}(\mathfrak g)## is a representation on the Lie algebra. Now, one often encounters the following formula for the curvature of the connection ##\omega##:
$$\Omega = D\omega = d\omega + \frac{1}2 [\omega, \omega].$$
However, if we use the representation ##\text{ad}:\mathfrak g \to \mathfrak{gl}(\mathfrak g)##, then the covariant exterior derivative of ##\omega## gives
$$D\alpha = d \alpha + \rho(\omega) \wedge \alpha = d \alpha + [\xi_k, \xi_l] \omega^k \wedge \omega^l = d\alpha + [\omega, \omega]$$.
But where have the factor ##1/2## gone?
I suspect my error might lie in one of the following:
(1): my definition of ##[\omega, \omega]## as ##[\xi_k, \xi_l] \omega^k \wedge \omega^l##.
(2): or, that what is meant by the expression ##\rho(\omega) \wedge \omega## is perhaps not ##[\xi_k, \xi_l] \omega^k \wedge \omega^l##.
But which one is it? And why?