AI 中文总结
该研究引入U(gl(N))及其q变形生成元的偏导数类似物，利用量子偏导数构建微分代数、定义量子德拉姆算子及莱布尼茨法则，还通过凯莱-哈密顿恒等式类似物构造投射模并定义陈类，为非交换背景下特征类研究提供新方法。
AI 中文摘要
我们引入了关于包络代数U(gl(N))及其q变形（即所谓的修正反射方程代数）生成元的偏导数类似物。利用这些量子偏导数，我们引入了相应的微分代数，定义了量子德拉姆算子，并展示了其作用的莱布尼茨法则。我们的莱布尼茨法则不同于经典类似物，源于量子对偶的构造。在代数U(gl(N))的情况下，我们给出了莱布尼茨法则的另一种形式，便于将微积分扩展到该代数的某些扩展。然后，我们利用U(gl(N))和反射方程代数生成矩阵的凯莱-哈密顿恒等式类似物来构造这些代数上的投射模。对于这些投射模，我们引入格拉斯曼联络并定义相应的陈类。
英文摘要
We introduce analogues of partial derivatives with respect to the generators of the enveloping algebras U(gl(N)) and their q-deformations, the so-called modified reflection equation algebras. Using these quantum partial derivatives, we introduce the corresponding differential algebras, define the quantum de Rham operator, and exhibit the Leibniz rule for its action. Our Leibniz rule differs from its classical analogue and stems from the construction of a quantum double. In the case of the algebra U(gl(N)), we present another form of the Leibniz rule that is more convenient for prolonging the differential calculus to certain extensions of this algebra. We then use analogues of the Cayley-Hamilton identity for the generating matrices of U(gl(N)) and of reflection equation algebras to construct projective modules over these algebras. For these projective modules, we introduce Grassmannian connections and define the corresponding Chern classes.
