Braiding and exponentiating noncommutative vector fields

量子演算：非交换几何中的微分形式与向量场

Quantum Calculi: differential forms and vector fields in noncommutative geometry

本文修订并扩展非交换向量场框架，填补旧有不足，阐明嘉当对与一阶微分的对应，并显式构造右/左泛向量场双模。

Almost commutative Riemannian geometry: wave operators

Associated to any (pseudo)-Riemannian manifold $M$ of dimension $n$ is an $n+1$-dimensional noncommutative differential structure $(Ω^1,\extd)$ on the manifold, with the extra dimension encoding the classical Laplacian as a noncommutative `vector field'. We use the classical connection, Ricci tensor and Hodge Laplacian to construct $(Ω^2,\extd)$ and a natural noncommutative torsion free connection $(\nabla,σ)$ on $Ω^1$. We show that its generalised braiding $σ:Ω^1\tensΩ^1\to Ω^1\tensΩ^1$ obeys the quantum Yang-Baxter or braid relations only when the original $M$ is flat, i.e their failure is g…

非交换几何中的Cartan结构方程与Levi-Civita联络

Cartan structure equations and Levi-Civita connection in noncommutative geometry

研究带有三角Hopf代数作用且非交换性与相关辫子兼容的代数上的微分和黎曼几何，通过扩展Cartan演算证明了曲率和挠率的向量场与形式方法的等价性，推导了Cartan结构方程和Bianchi恒等式，并利用Koszul公式证明了任意伪黎曼度量的Levi-Civita联络的存在唯一性。