On the geometry of $(\sigma,\tau)$-algebras

TL;DR

This paper introduces a new framework called $(\sigma, au)$-algebras for twisted differential calculus applicable to both noncommutative and commutative algebras, extending concepts like connections, torsion, and curvature.

Contribution

It develops the theory of $(\sigma, au)$-algebras, including modules, connections, and geometric notions such as Levi-Civita connections, with detailed examples over matrix algebras.

Findings

Existence of $(\sigma, au)$-connections on projective modules

Definition of torsion, curvature, and hermitian compatibility for $(\sigma, au)$-connections

Explicit examples over matrix algebras

Abstract

We introduce $(\sigma,\tau)$-algebras as a framework for twisted differential calculi over noncommutative, as well as commutative, algebras with motivations from the theory of $\sigma$-derivations and quantum groups. A $(\sigma,\tau)$-algebra consists of an associative algebra together with a set of $(\sigma,\tau)$-derivations, and corresponding notions of $(\sigma,\tau)$-modules and connections are introduced. We prove that $(\sigma,\tau)$-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 $(\sigma,\tau)$-connection. To illustrate the novel concepts, we consider $(\sigma,\tau)$-algebras and connections over matrix algebras in detail.