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.