  We develop new mathematical tools for the study of the double copy and
colour-kinematics duality for tree-level scattering amplitudes using the
properties of Lie polynomials. We show that the $S$-map that was defined to
simplify super-Yang--Mills multiparticle superfields is in fact a new Lie
bracket on the dual space of Lie polynomials. We introduce {\it Lie polynomial
currents} based on Berends-Giele recursion for biadjoint scalar tree amplitudes
that take values in Lie polynomials. Field theory amplitudes are obtained from
the Lie polynomial amplitudes by numerators characterized as homomorphisms from
the free Lie algebra to kinematic data. Examples are presented for the
biadjoint scalar, Yang--Mills theory and the nonlinear sigma model. That these
theories satisfy the Bern-Carrasco-Johansson amplitude relations follows from
the identities we prove for the Lie polynomial amplitudes and the existence of
BCJ numerators.
  A KLT map from Lie polynomials to their dual is obtained by nesting the S-map
Lie bracket; the matrix elements of this map yield a recently proposed
`generalized KLT matrix', and this reduces to the usual KLT matrix when its
entries are restricted to a basis. Using this, we give an algebraic proof for
the cancellation of double poles in the KLT formula for gravity amplitudes. We
finish with some remarks on numerators and colour-kinematics duality from this
perspective.
