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

By J. Arnlind, Kwalombota Ilwale. 2022

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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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/193e38026ac362dee46117f926dbeb3fed5f56fc