Nonassociative geometry in quasi-Hopf representation categories II: Connections and curvature

TL;DR

This paper develops a framework for noncommutative and nonassociative differential geometry within quasi-Hopf algebra representation categories, focusing on connections, curvature, and potential applications to nonassociative gravity theories.

Contribution

It introduces categorical constructions for connections and curvature in nonassociative geometry, extending previous work to include tensor products and internal homs.

Findings

Constructed morphisms for lifting connections to tensor products and internal homs.

Described curvature within the nonassociative geometric framework.

Formulated Einstein-Cartan geometry for nonassociative gravity theories.

Abstract

We continue our systematic development of noncommutative and nonassociative differential geometry internal to the representation category of a quasitriangular quasi-Hopf algebra. We describe derivations, differential operators, differential calculi and connections using universal categorical constructions to capture algebraic properties such as Leibniz rules. Our main result is the construction of morphisms which provide prescriptions for lifting connections to tensor products and to internal homomorphisms. We describe the curvatures of connections within our formalism, and also the formulation of Einstein-Cartan geometry as a putative framework for a nonassociative theory of gravity.