AI 中文总结
本文研究带有三角Hopf代数表示的非对易代数的微分几何，提出任意联络比安基恒等式的新方法，证明其等价性并得到广义相对论比安基恒等式的非对易类似物。
AI 中文摘要
我们首先介绍带有三角Hopf代数表示的非对易代数的微分几何，其非对易性由通用R矩阵编码，微分几何由这些数据典范地构造。接着，我们针对任意联络（不一定是双模联络）的曲率和挠率的比安基恒等式发展了一种新方法，利用联络的嘉当演算，证明了其在外微分形式与张量场表述下的全局形式等价，特别得到了广义相对论中熟知的第一、第二比安基恒等式的非对易类似物。
英文摘要
We first present an introduction to the differential geometry of noncommutative algebras that carry a representation of a triangular Hopf algebra. Their noncommutativity is encoded in the universal R-matrix. The differential geometry is canonically constructed from these data. We then develop a new approach to the Bianchi identities for curvature and torsion of arbitrary connections (not necessarily bimodule connections). Using the Cartan calculus for connections, we prove the equivalence of their global formulations in terms of exterior forms and tensor fields. In particular, we obtain the noncommutative analogues of the first and second Bianchi identities familiar from general relativity.
Comments22 pages
