A Lie bracket for the momentum kernel

Understand

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.