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

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.

v1: 56 pages. v2: 57 pages. Introduction expanded. Published in Journal of Geometry and Physics

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• Finite Gauge Transformations and Geometry in Double Field Theory
• Twisted Poisson Structures and Non-commutative/non-associative Closed String Geometry
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Cited by in corpus (8)

• $G_2$-structures and quantization of non-geometric M-theory backgrounds
• Braided $L_{\infty }$-Algebras, Braided Field Theory and Noncommutative Gravity
• Braided Symmetries in Noncommutative Field Theory
• Nonassociative geometry of nonholonomic phase spaces with star R-flux string deformations and (non) symmetric metrics
• Homotopy Lie Algebras of Gravity and their Braided Deformations
• Deformation quantization of principal bundles
• Extended Field Theories as higher Kaluza-Klein theories
• Slack Hopf Monads