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

Barnes, Gwendolyn E. and Schenkel, Alexander and Szabo, Richard J. (2015) Nonassociative geometry in quasi-Hopf representation categories I: bimodules and their internal homomorphisms. Journal of Geometry and Physics, 89 . pp. 111-152. ISSN 0393-0440

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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.

Item Type: Article
Keywords: Noncommutative/nonassociative differential geometry; Quasi-Hopf algebras; Braided monoidal categories; Internal homomorphisms; Cochain twist quantization
Schools/Departments: University of Nottingham, UK > Faculty of Science > School of Mathematical Sciences
Identification Number: 10.1016/j.geomphys.2014.12.005
Depositing User: Schenkel, Dr Alexander
Date Deposited: 02 Mar 2017 15:28
Last Modified: 28 Mar 2017 10:56

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