Structure constants of SU(N)

  • Thread starter Lester
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  • #1
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Hi there,

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

[tex]\sum_a f_{abc}f_{ade}[/tex]

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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  • #2
samalkhaiat
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Hi there,

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

[tex]\sum_a f_{abc}f_{ade}[/tex]

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 [itex]d_{abc}[/itex] which vanish in SU(2);

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

Another useful identities are (Jacobi identities)

[tex]f_{abe}d_{ecd} + f_{cbe}d_{aed} + f_{dbe}d_{ace} = 0[/tex]

and the usual one for the structure constants [itex]f_{abc}[/itex]


regards

sam
 
  • #3
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Thanks a lot Sam. This was the formula I was looking for.

Jon
 
  • #4
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.

Thanks in advance
 
  • #5
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:

[tex]d^{ace}d^{bde}+d^{ade}d^{bce}+d^{bae}d^{cde}=0[/tex]

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

Thanks.
 

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