A bstractOne important discovery in recent years is that the total amplitude of gauge theory can be written as BCJ form where kinematic numerators satisfy Jacobi identity. Although the existence of such kinematic numerators is no doubt, the simple and explicit construction is still an important problem. As a small step, in this note we provide an algebraic approach to construct these kinematic numerators. Under our Feynman-diagram-like construction, the Jacobi identity is manifestly satisfied. The corresponding color ordered amplitudes satisfy off-shell KK-relation and off-shell BCJ relation similar to the color ordered scalar theory. Using our construction, the dual DDM form is also established.

中文翻译：

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BCJ 分子的代数方法

摘要 近年来的一个重要发现是规范理论的总振幅可以写成运动学分子满足雅可比恒等式的 BCJ 形式。虽然这种运动学分子的存在是毋庸置疑的，但简单明确的构造仍然是一个重要的问题。作为一小步，在本笔记中，我们提供了一种代数方法来构造这些运动学分子。在我们的费曼图式构造下，雅可比恒等式显然得到满足。相应的色序幅度满足类似于色序标量理论的壳外 KK 关系和壳外 BCJ 关系。使用我们的构造，也建立了双 DDM 形式。