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

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

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20 Citations (Scopus)
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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 languageEnglish
Pages (from-to)111-152
Number of pages42
JournalJournal of Geometry and Physics
Volume89
DOIs
Publication statusPublished - Mar 2015

Keywords

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

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