AI 中文总结
本文在辫子幺半范畴中构造辫子Hopf代数胚，引入辫子Lie-Rinehart代数，并证明其右不变向量场与量子主丛上的等变向量场同构，且建立泛包络代数与jet空间的配对。
AI 中文摘要
我们在域$k$上的辫子幺半范畴中构造辫子Hopf代数胚，并利用图形表示研究其性质。然后，我们研究与辫子Hopf Galois扩张相关的Ehresmann-Schauenburg Hopf代数胚。我们引入辫子Lie-Rinehart代数，它推广了\cite{ALP24, ALP23, ALP25}中的定义，并在对称条件下研究其泛包络代数。特别地，在三角Hopf代数的模范畴中，我们引入与辫子Hopf代数胚相关的Lie-Rinehart代数（辫子Lie代数胚）。随后，我们聚焦于辫子Ehresmann-Schauenburg Hopf代数胚的情形，研究其右不变向量场，结果表明这些向量场作为Lie-Rinehart代数与量子主丛（Hopf Galois扩张）上的$H$-等变向量场同构。最后，在源子代数和目标子代数属于辫子中心的情形下，我们引入与辫子Hopf代数胚相关的辫子jet Hopf代数胚。此外，我们证明了辫子右不变向量场的$k$阶泛包络代数与辫子Hopf代数胚的第$k$个jet空间之间存在对偶配对，并且在有限性条件下，其辫子右不变向量场的泛包络代数与其jet Hopf代数胚之间存在斜配对。
英文摘要
We construct braided Hopf algebroids in a braided monoidal category over a field $k$ and study its properties using graphical representation. Then we study Ehresmann-Schauenburg Hopf algebroids assocaited to braided Hopf Galois extensions. We introduce braided Lie-Rinehart algebras which generalize the definition in \cite{ALP24, ALP23, ALP25} and study its universal enveloping algebra under symmetric condition. In particular, we introduce Lie-Rinehart algebras (braided Lie-algebroids) associated with braided Hopf algebroids in the category of the module of a triangular Hopf algebra. We then focus on the case for braided Ehresmann-Schauenburg Hopf algebroids and study their right invariant vector fields, which turn out to be isomorphic to the $H$-equivariant vector fields on the quantum principal bundle (Hopf Galois extension) as Lie-Rinehart algebras. Finally, we introduce braided jet Hopf algebroids associate to braided Hopf algebroids in the case that the source and target subalgebra belong to the braided center. Moreover, we show there is a dual pairing between the $k$-order universal enveloping algebra of the braided right invariant vector fields and the $k$-th jet space of a braided Hopf algebroid and further more a skew pairing between the universal enveloping algebra of its braided right invariant vector fields and its jet Hopf algebroid under a finiteness condition.
