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On the geometry of $(σ,τ)$-algebras

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.