- #1
wac03
- 6
- 0
Dear friends
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 algebra
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.
thank you in advance wissam
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
Code:
[ 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
Code:
[ tex ][D_{m},D_{n}]F_{ab}=[F_{mn},F_{ab}][ /tex ]
thank you in advance wissam