Nonassociative geometry in quasi-Hopf representation categories I: Bimodules and their internal homomorphisms

TL;DR

This paper develops a framework for studying noncommutative and nonassociative algebras and bimodules within quasi-Hopf algebra representation categories, introducing internal homomorphisms and tensor products, with applications to deformation quantization and noncommutative geometry.

Contribution

It introduces internal homomorphisms and tensor products for bimodules in quasi-Hopf categories, enabling systematic development of noncommutative differential geometry.

Findings

Explicit description of evaluation and composition morphisms for internal homs

Construction of a braided closed monoidal category of symmetric bimodules

Application to deformation quantization of vector bundles

Abstract

We systematically study noncommutative and nonassociative algebras A and their bimodules as algebras and bimodules internal to the representation category of a quasitriangular quasi-Hopf algebra. We enlarge the morphisms of the monoidal category of A-bimodules by internal homomorphisms, and describe explicitly their evaluation and composition morphisms. For braided commutative algebras A the full subcategory of symmetric A-bimodule objects is a braided closed monoidal category, from which we obtain an internal tensor product operation on internal homomorphisms. We describe how these structures deform under cochain twisting of the quasi-Hopf algebra, and apply the formalism to the example of deformation quantization of equivariant vector bundles over a smooth manifold. Our constructions set up the basic ingredients for the systematic development of differential geometry internal to the quasi-Hopf representation category, which will be tackled in the sequels to this paper, together with applications to models of noncommutative and nonassociative gravity such as those anticipated from non-geometric string theory.