TY - JOUR

T1 - Nonassociative geometry in quasi-Hopf representation categories I

T2 - Bimodules and their internal homomorphisms

AU - Barnes, Gwendolyn E.

AU - Schenkel, Alexander

AU - Szabo, Richard J.

N1 - "G.E.B. is a Commonwealth Scholar, funded by the UK Government. The work of A.S. is supported by a Research Fellowship of the Deutsche Forschungsgemeinschaft (DFG). The work of R.J.S. is supported in part by the Consolidated Grant ST/J000310/1 from the UK Science and Technology Facilities Council."

PY - 2015/3

Y1 - 2015/3

N2 - 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.

AB - 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.

KW - Braided monoidal categories

KW - Cochain twist quantization

KW - Internal homomorphisms

KW - Noncommutative/nonassociative differential geometry

KW - Quasi-Hopf algebras

U2 - 10.1016/j.geomphys.2014.12.005

DO - 10.1016/j.geomphys.2014.12.005

M3 - Article

AN - SCOPUS:84923018013

VL - 89

SP - 111

EP - 152

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

ER -