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

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.