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

Understand

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.