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

Barnes, Gwendolyn E.; Schenkel, Alexander; Szabo, Richard J.

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.

Citation

Barnes, G. E., Schenkel, A., & Szabo, R. J. (2015). Nonassociative geometry in quasi-Hopf representation categories I: bimodules and their internal homomorphisms. Journal of Geometry and Physics, 89, 111-152. https://doi.org/10.1016/j.geomphys.2014.12.005

• Journal Article Type: Article
• Acceptance Date: Dec 8, 2014
• Online Publication Date: Dec 17, 2014
• Publication Date: 2015-03
• Deposit Date: Mar 2, 2017
• Publicly Available Date: Mar 2, 2017
• Journal: Journal of Geometry and Physics
• Print ISSN: 0393-0440
• Electronic ISSN: 0393-0440
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 89
• Pages: 111-152
• DOI: https://doi.org/10.1016/j.geomphys.2014.12.005
• Keywords: Noncommutative/nonassociative differential geometry; Quasi-Hopf algebras; Braided monoidal categories; Internal homomorphisms; Cochain twist quantization
• Public URL: https://nottingham-repository.worktribe.com/output/746275
• Publisher URL: https://doi.org/10.1016/j.geomphys.2014.12.005
• Related Public URLs: https://arxiv.org/abs/1409.6331
• Additional Information: This article is maintained by: Elsevier; Article Title: Nonassociative geometry in quasi-Hopf representation categories I: Bimodules and their internal homomorphisms; Journal Title: Journal of Geometry and Physics; CrossRef DOI link to publisher maintained version: https://doi.org/10.1016/j.geomphys.2014.12.005; Content Type: article; Copyright: Copyright © 2014 Elsevier B.V. All rights reserved.
• Contract Date: Mar 2, 2017