# Completeness relation for SO(N)

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

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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!

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.