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

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Nonabelian gauge field

Loading...

Similar Threads for Nonabelian gauge field | Date |
---|---|

A Pushforward of Smooth Vector Fields | Jan 12, 2018 |

Temporal Gauge | Mar 21, 2012 |

Christoffel Symbols - Gauge Fields | Jun 23, 2011 |

Covariant derivative vs Gauge Covariant derivative | Jun 16, 2011 |

Gauge covariant derivative in curvilinear coordinates | Apr 30, 2007 |

**Physics Forums - The Fusion of Science and Community**