# Structure constants of SU(N)

## Main Question or Discussion Point

Hi there,

Does anybody know how to exploit the product of structure constants of SU(N) through Kroenecker deltas? I mean

$$\sum_a f_{abc}f_{ade}$$

I know this for SU(2) as in this case I have the Levi-Civita symbol but in other cases I was not able to recover it in literature. Any help appreciated.

Jon

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samalkhaiat
Hi there,

Does anybody know how to exploit the product of structure constants of SU(N) through Kroenecker deltas? I mean

$$\sum_a f_{abc}f_{ade}$$

I know this for SU(2) as in this case I have the Levi-Civita symbol but in other cases I was not able to recover it in literature. Any help appreciated.

Jon
It is given in terms of the totally symmetric coefficients $d_{abc}$ which vanish in SU(2);

$$f_{abe}f_{cde} = \frac{2}{n} (\delta_{ac}\delta_{bd} - \delta_{ad}\delta_{bc}) + ( d_{ace}d_{bde} - d_{bce}d_{ade})$$

Another useful identities are (Jacobi identities)

$$f_{abe}d_{ecd} + f_{cbe}d_{aed} + f_{dbe}d_{ace} = 0$$

and the usual one for the structure constants $f_{abc}$

regards

sam

Thanks a lot Sam. This was the formula I was looking for.

Jon

Dear Sam & Lester

Can you please tell me any book reference for these identities involving $d^{abc}$ in jacobi identity and the structure constants contraction.

Dear samalkhaiat

Your formula for jacobi identity is wrong. Actually it should have all d^{abc} instead of the f^{abc} everywhere. The correct Jacobi identity is:

$$d^{ace}d^{bde}+d^{ade}d^{bce}+d^{bae}d^{cde}=0$$

Kindly provide me any references for contraction formula for structure constants of SU(N).

Thanks.