  We study the Kawai-Lewellen-Tye (KLT) relations for quantum field theory by
reformulating it as an isomorphism between two Lie algebras. We also show how
explicit formulas for KLT relations arise when studying rational functions on
${\mathcal M}_{0,n}$, and prove identities that allow for arbitrary rational
functions to be expanded in any given basis. Via the Cachazo-He-Yuan formulas
for, these identities also lead to new formulas for gauge and gravity tree
amplitudes, including formulas for so-called Bern-Carrasco-Johansson
numerators, in the case of non-linear sigma model and
maximal-helicity-violating Yang-Mills amplitudes.
