Completeness relation for SO(N)

[Einj]

Hello everyone,
I was wondering if anyone knows what the completeness relation for the fundamental representation of SO(N) is.
For example, in the SU(N) we know that, if T^a_{ij} are the generators of the fundamental representation then we have the following relation
$$
T^a_{ij}T^a_{km}=\frac{1}{2}\left(\delta_{im}\delta_{jk}-\frac{1}{N}\delta_{ij}\delta_{km}\right)
$$
This follows from the fact that the T^a, together with the identity form a complete basis for the N\times N complex matrices.

Does anyone know how to find the analogous for SO(N) (if any)?

Thanks a lot!

 

[jambaugh]

There is not an analogue. See http://repositorio.unesp.br/bitstream/handle/11449/23433/WOS000234099500010.pdf?sequence=1 by C. C. Nishi
equation 22 and the paragraph following it.

 

[Supper]

Hello everyone,
I was wondering if anyone knows what the completeness relation for the fundamental representation of SO(N) is.
For example, in the SU(N) we know that, if T^a_{ij} are the generators of the fundamental representation then we have the following relation
$$
T^a_{ij}T^a_{km}=\frac{1}{2}\left(\delta_{im}\delta_{jk}-\frac{1}{N}\delta_{ij}\delta_{km}\right)
$$
This follows from the fact that the T^a, together with the identity form a complete basis for the N\times N complex matrices.

Does anyone know how to find the analogous for SO(N) (if any)?

Thanks a lot!

Such a relation you can obtain in fundamental rep by yourself just using definition of generators
$$
(\lambda_{ab})_{cd}=-i(\delta_{ac}\delta_{bd}-\delta_{ad}\delta_{bc}) \in so(n).
$$
By contracting with an other $\lambda_{ab})_{$ you get
$$
(\lambda_{ab})_{cd}(\lambda_{ab})_{ef}=-2(\delta_{ce}\delta_{df}-\delta_{cf}\delta_{de}).
$$

In spinor rep it's going to be more complicated.