Dear friends(adsbygoogle = window.adsbygoogle || []).push({});

I am here with mathematical physics question:

we know tha if i have a compact Lie group G with g its Lie algebra, and a connection A on the fibre,

For nonabelain Lie algebra

The relation between covariant derivative and the curvature of A is

for any representation of g the Lie algebraCode (Text):[ tex ]\begin{equation*}[D_{m},D_{n}]F_{ab}=[F_{mn},F_{ab}]\end{equation*}[ /tex ]

with

D is the covariant derivative

F the curvature of the connection A

my problem:

I will be so grateful if someone could help me to prove that

is valid for any representation of the Lie algebra g especially for the fundamental (defining) representation, because i already did it for the adjoint representation of g.Code (Text):[ tex ][D_{m},D_{n}]F_{ab}=[F_{mn},F_{ab}][ /tex ]

thank you in advance wissam

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# Nonabelian gauge field

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