动量核的一个李括号
A Lie bracket for the momentum kernel
AI总结：
本研究利用李多项式性质构建树级散射振幅研究的新数学工具，证明S映射是李多项式对偶空间的新李括号，提出李多项式流，推导相关振幅关系，构造广义KLT矩阵并证明引力振幅KLT公式双极点抵消，深化色-运动学对偶性研究。
摘要：
我们利用李多项式的性质，开发了用于研究树级散射振幅的双重拷贝和色-运动学对偶性的新数学工具。我们证明，为简化超杨-米尔斯多粒子超场而定义的S映射，实际上是李多项式对偶空间上的一个新的李括号。我们针对取值于李多项式的双伴随标量树振幅，提出了基于Berends-Giele递推的“李多项式流”。场论振幅可通过李多项式振幅得到，其分子被刻画为从自由李代数到运动学数据的同态。我们给出了双伴随标量理论、Yang-Mills理论和非线性σ模型的示例。这些理论满足Bern-Carrasco-Johansson（BCJ）振幅关系，可由我们证明的李多项式振幅恒等式以及BCJ分子的存在性推导得出。通过嵌套S映射李括号，我们得到了从李多项式到其对偶空间的KLT映射；该映射的矩阵元对应于近期提出的“广义KLT矩阵”，当将其元素限制在一组基上时，它会退化为通常的KLT矩阵。利用这一结果，我们对引力振幅KLT公式中双极点的抵消给出了代数证明。最后，我们从这一视角对分子和色-运动学对偶性做了一些评述。
英文摘要：
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.
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