A bstractIn this note we provide a new construction of BCJ dual-trace factor using the kinematic algebra proposed in arXiv:1105.2565 and arXiv:1212.6168. Different from the construction given in arXiv:1304.2978 based on the proposal of arXiv:1103.0312, the method used in this note exploits the adjoint representation of kinematic algebra and the use of inner product in dual space. The dual-trace factor defined in this way naturally satisfies cyclic symmetry condition but not KK-relation, just like the trace of U(N) Lie algebra satisfies cyclic symmetry condition, but not KK-relation. In other words the new construction naturally leads to formulation sharing more similarities with the color decomposition of Yang-Mills amplitude.

中文翻译：

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Yang-Mills理论中双迹因子的构建注意事项

摘要在本笔记中，我们使用 arXiv:1105.2565 和 arXiv:1212.6168 中提出的运动代数提供了一种新的 BCJ 双迹因子构造。与基于 arXiv:1103.0312 提议的 arXiv:1304.2978 中给出的构造不同，本笔记中使用的方法利用了运动代数的伴随表示和对偶空间中内积的使用。这样定义的对偶迹因子自然满足循环对称条件但不满足KK-关系，就像U(N)李代数的迹满足循环对称条件但不满足KK-关系。换句话说，新构造自然会导致公式与杨-米尔斯振幅的颜色分解有更多相似之处。