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arXiv 2207.08400math.QAmath-phmath.MP
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关于$(σ,τ)$ -代数的几何

On the geometry of $(σ,τ)$-algebras

Joakim Arnlind, Kwalombota Ilwale

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摘要：

我们引入$(σ,τ)$ -代数作为非交换和交换代数上的扭曲微分演算的框架，其动机来自$σ$ -导子和量子群理论。一个$(σ,τ)$ -代数由一个结合代数和一组$(σ,τ)$ -导子组成，并引入了相应的$(σ,τ)$ -模和联络的概念.我们证明了$(σ,τ)$ -联络存在于投射模上，并引入了挠和曲率的概念，以及与厄米特形式的相容性，从而定义了Levi-Civita $(σ,τ)$ -联络.为了说明新的概念，我们考虑$(σ,τ)$ -代数和连接矩阵代数的详细。

英文摘要：

We introduce $(σ,τ)$-algebras as a framework for twisted differential calculi over noncommutative, as well as commutative, algebras with motivations from the theory of $σ$-derivations and quantum groups. A $(σ,τ)$-algebra consists of an associative algebra together with a set of $(σ,τ)$-derivations, and corresponding notions of $(σ,τ)$-modules and connections are introduced. We prove that $(σ,τ)$-connections exist on projective modules, and introduce notions of both torsion and curvature, as well as compatibility with a hermitian form, leading to the definition of a Levi-Civita $(σ,τ)$-connection. To illustrate the novel concepts, we consider $(σ,τ)$-algebras and connections over matrix algebras in detail.

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