Viewing the space of cotraces in the structural coalgebra of a principal coaction as a noncommutative counterpart of the classical Cartan model, we construct the cyclic-homology Chern–Weil homomorphism. To realize the thus constructed Chern–Weil homomorphism as a Cartan model of the homomorphism tautologically induced by the classifying map on cohomology, we replace the unital subalgebra of coaction-invariants by its natural H-unital nilpotent extension (row extension). Although the row-extension algebra provides a drastically different model of the cyclic object, we prove that, for any row extension of any unital algebra over a commutative ring, the row-extension Hochschild complex and the usual Hochschild complex are chain homotopy equivalent. It is the discovery of an explicit homotopy formula that allows us to improve the homological quasi-isomorphism arguments of Loday and Wodzicki. We work with families of principal coactions, and instantiate our noncommutative Chern–Weil theory by computing the cotrace space and analyzing a dimension-drop-like effect in the spirit of Feng and Tsygan for the quantum-deformation family of the standard quantum Hopf fibrations.

观察作为主要协同作用的结构共轭结构中共迹线的空间，作为经典Cartan模型的非可交换对等体，我们构造了循环同调Chern-Weil同态。为了将这样构造的Chern-Weil同态实现为由同构分类图重言式同构的Cartan模型，我们用自然H单位幂幂扩展（行扩展）代替了coact-不变式的统一子代数。尽管行扩展代数提供了循环对象的完全不同的模型，但我们证明，对于交换环上任何单位代数的任何行扩展，行扩展Hochschild复数和通常的Hochschild复数都是链同伦等效的。这是一个明确的同伦公式的发现，它使我们能够改进Loday和Wodzicki的同构准同构论点。我们处理主要相互作用的族，并通过计算同迹空间并分析Feng和Tsygan精神下的标准量子Hopf纤维的量子形变族来实例化非交换Chern-Weil理论。