= SU(N) fundamental completeness relation
{c}
{title2=$\sum_a T^a_{ij}T^a_{kl}=\tfrac12(\delta_{il}\delta_{jk}-\delta_{ij}\delta_{kl}/N)$}
For traceless <Hermitian matrices> representing the <Lie algebra> of the <special unitary group> $SU(N)$, normalized by $\operatorname{tr}(T^aT^b)=\delta^{ab}/2$, completeness of $\{I/\sqrt{2N},T^a\}$ in the space of <Hermitian matrices> gives the displayed identity. In particular $\sum_a T^aXT^a=\tfrac12\operatorname{tr}(X)I-X/(2N)$. For traceless $X=T^b$, this becomes $\sum_a T^aT^bT^a=-T^b/(2N)$. These identities reduce <QCD> <color charge> sums without choosing explicit matrices.
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