  We find simple expressions for the kinematic numerators of one-loop MHV
amplitudes in maximally supersymmetric Yang-Mills theory and supergravity, for
any multiplicity. The gauge theory numerators satisfy the
Bern-Carrasco-Johansson (BCJ) duality between color and kinematics, so that the
gravity numerators are simply the square of the gauge theory ones. The duality
holds because the numerators can be written in terms of structure constants of
a kinematic algebra, which is familiar from the BCJ organization of self-dual
gauge theory and gravity. The close connection that we find between one-loop
amplitudes in the self-dual case and in the maximally supersymmetric case is
the converse to the dimension-shifting formula. The starting point for arriving
at our expressions is the dimensional reduction of ten-dimensional amplitudes
obtained in the field-theory limit of open superstrings.
