### 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.

Original language | English |
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Pages (from-to) | 111-152 |

Number of pages | 42 |

Journal | Journal of Geometry and Physics |

Volume | 89 |

DOIs | |

Publication status | Published - Mar 2015 |

### Keywords

- Braided monoidal categories
- Cochain twist quantization
- Internal homomorphisms
- Noncommutative/nonassociative differential geometry
- Quasi-Hopf algebras

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## Cite this

*Journal of Geometry and Physics*,

*89*, 111-152. https://doi.org/10.1016/j.geomphys.2014.12.005