Nonassociative geometry in quasi-Hopf representation categories I: bimodules and their internal homomorphisms
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...
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| Format: | Article |
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Elsevier
2015
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| Online Access: | https://eprints.nottingham.ac.uk/41007/ |
| _version_ | 1848796176537616384 |
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| author | Barnes, Gwendolyn E. Schenkel, Alexander Szabo, Richard J. |
| author_facet | Barnes, Gwendolyn E. Schenkel, Alexander Szabo, Richard J. |
| author_sort | Barnes, Gwendolyn E. |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T19:43:49Z |
| format | Article |
| id | nottingham-41007 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:43:49Z |
| publishDate | 2015 |
| publisher | Elsevier |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-410072020-05-04T17:03:36Z https://eprints.nottingham.ac.uk/41007/ Nonassociative geometry in quasi-Hopf representation categories I: bimodules and their internal homomorphisms Barnes, Gwendolyn E. Schenkel, Alexander Szabo, Richard J. 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. Elsevier 2015-03-31 Article PeerReviewed Barnes, Gwendolyn E., 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 Noncommutative/nonassociative differential geometry; Quasi-Hopf algebras; Braided monoidal categories; Internal homomorphisms; Cochain twist quantization https://doi.org/10.1016/j.geomphys.2014.12.005 doi:10.1016/j.geomphys.2014.12.005 doi:10.1016/j.geomphys.2014.12.005 |
| spellingShingle | Noncommutative/nonassociative differential geometry; Quasi-Hopf algebras; Braided monoidal categories; Internal homomorphisms; Cochain twist quantization Barnes, Gwendolyn E. Schenkel, Alexander Szabo, Richard J. Nonassociative geometry in quasi-Hopf representation categories I: bimodules and their internal homomorphisms |
| title | Nonassociative geometry in quasi-Hopf representation categories I: bimodules and their internal homomorphisms |
| title_full | Nonassociative geometry in quasi-Hopf representation categories I: bimodules and their internal homomorphisms |
| title_fullStr | Nonassociative geometry in quasi-Hopf representation categories I: bimodules and their internal homomorphisms |
| title_full_unstemmed | Nonassociative geometry in quasi-Hopf representation categories I: bimodules and their internal homomorphisms |
| title_short | Nonassociative geometry in quasi-Hopf representation categories I: bimodules and their internal homomorphisms |
| title_sort | nonassociative geometry in quasi-hopf representation categories i: bimodules and their internal homomorphisms |
| topic | Noncommutative/nonassociative differential geometry; Quasi-Hopf algebras; Braided monoidal categories; Internal homomorphisms; Cochain twist quantization |
| url | https://eprints.nottingham.ac.uk/41007/ https://eprints.nottingham.ac.uk/41007/ https://eprints.nottingham.ac.uk/41007/ |