标题： 动量核的李括号 显示英文标题

标题： A Lie bracket for the momentum kernel

摘要： 我们发展了新的数学工具，用于利用李多项式的性质研究树级散射振幅的双拷贝和颜色-运动学对偶。 我们证明了定义用来简化超杨-米尔斯多粒子超场的$S$-映射实际上是在李多项式对偶空间上的新李括号。 我们基于双仿射标量树振幅的Berends-Giele递归引入{\it 李多项式电流}，这些振幅取值于李多项式。 通过数值上刻画为从自由李代数到运动学数据的同态，可以从李多项式振幅得到场论振幅。 给出了双仿射标量、杨-米尔斯理论以及非线性sigma模型的例子。 这些理论满足Bern-Carrasco-Johansson振幅关系，这是由我们对李多项式振幅证明的恒等式以及BCJ数值的存在性所推导出的。 通过嵌套S-映射李括号，可以从李多项式到其对偶空间获得一个KLT映射；此映射的矩阵元给出了最近提出的‘广义KLT矩阵’，当其条目限制为基底时，该矩阵退化为通常的KLT矩阵。 利用这一点，我们给出了引力振幅KLT公式中二阶极点抵消的一个代数证明。 最后，我们从这个视角对数值和颜色-运动学对偶作了一些评论。 显示英文摘要

摘要： 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.