In the formulation of connections on principal bundles, one derives an(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Factor 1/2 in the Curvature Two-form of a Connection Principal Bundle

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