论文标题
无限空间上的非交通差几何形状
Noncommutative Differential Geometry on Infinitesimal Spaces
论文作者
论文摘要
在本文中，我们使用非共同差异几何形状的语言来形式化离散的微积分。我们首先简要审查POSET的反限制，作为拓扑空间的近似值。然后，我们展示如何将$ c^*$ - 代数与poset相关联，从而使其具有分段线性结构。此外，我们解释了在歧管$ m $上连续函数$ c（m）$的代数如何通过POSETS上的$ C^*$ - 代数的直接限制来近似。最后，本着非共同差异几何形状的精神，我们在每个poset上定义了有限的尺寸光谱三重。我们展示了如何用DIRAC运算符将通常的有限差分计算作为换向器的特征值。在$ \ mathbb {r}^d $中的$ d $ - lattice和torus $ \ mathbb {t}^d $的情况下，我们证明了融合结果。
In this paper, we use the language of noncommutative differential geometry to formalise discrete differential calculus. We begin with a brief review of inverse limit of posets as an approximation of topological spaces. We then show how to associate a $C^*$-algebra over a poset, giving it a piecewise-linear structure. Furthermore, we explain how dually the algebra of continuous function $C(M)$ over a manifold $M$ can be approximated by a direct limit of $C^*$-algebras over posets. Finally, in the spirit of noncommutative differential geometry, we define a finite dimensional spectral triple on each poset. We show how the usual finite difference calculus is recovered as the eigenvalues of the commutator with the Dirac operator. We prove a convergence result in the case of the $d$-lattice in $\mathbb{R}^d$ and for the torus $\mathbb{T}^d$.
