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

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

Authors

Gwendolyn E. Barnes

Richard J. Szabo



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.

Journal Article Type Article
Publication Date 2015-03
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
APA6 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
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
Publisher URL https://doi.org/10.1016/j.geomphys.2014.12.005
Related Public URLs https://arxiv.org/abs/1409.6331
Copyright Statement Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by-nc-nd/4.0
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.

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://creativecommons.org/licenses/by-nc-nd/4.0





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