Undergrad Covariant derivative of field strength tensor

[d8586]

Hi,

I am struggling to derive the relations on the right hand column of eq.(4) in https://arxiv.org/pdf/1008.4884.pdfEven the easy abelian one (third row)

which is

$$D_\rho B_{\mu\nu}=\partial_\rho B_{\mu\nu}$$

doesn't match my calculation

Since

$$D_\rho B_{\mu\nu}=(\partial_\rho+i g B_\rho)(\partial_\mu B_\nu-\partial_\nu B_\mu)$$

the equation seems to imply that

$$B_\rho (\partial_\mu B_\nu-\partial_\nu B_\mu)=0$$

What am I missing?

 

[Orodruin]

What am I missing?

The field tensor transforms according to the adjoint representation, not the fundamental one. The structure constants for an Abelian group are zero.

 

[d8586]

The field tensor transforms according to the adjoint representation, not the fundamental one. The structure constants for an Abelian group are zero.

Thank you very much!