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

Abstract

MSC:16T05 17B37 46L87 53D55Keywords: Noncommutative/nonassociative differential geometry Quasi-Hopf algebras Braided monoidal categories Internal homomorphisms Cochain twist quantization a b s t r a c t 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.

Perspectives on Nonassociative Geometry

“…Physically consistent models with novel properties in the context of quantum mechanics were constructed in [25] using this formalism, and of Euclidean scalar quantum field theory in [23]. To extend these considerations to more complicated field theories, a general systematic formalism was developed in [6,7] for differential geometry on noncommutative and nonassociative spaces internal to the representation category of any quasi-Hopf algebra, generalizing and extending earlier work [15,8,2]. This is the starting point for the present contribution.…”

“…The purpose of this contribution is to unpack and make explicit the somewhat abstract categorical constructions of [6,7] in a less formal language that we hope will be palatable to a larger audience. We focus on the special case of most physical relevance: the cochain twist quantization of a classical manifold; this construction is reviewed in Section 2.…”

“…We further restrict to trivial vector bundles over these noncommutative and nonassociative spaces with diagonal action of the pertinent Hopf algebra of symmetries of the non-geometric background. This simplification enables us to give very explicit "local" descriptions of the noncommutative and nonassociative geometry while still retaining generic features and indicating how the general formalism of [6,7] may be applied to constructions of physically viable field theories; in particular, we give concrete realizations of the pertinent bimodule operations for homomorphism bundles. In Section 3 we apply this framework to obtain explicit expressions for connections and their curvatures on noncommutative and nonassociative vector bundles.…”

“…E = M × C n → M with bundle projection given by projecting on the first factor. For a discussion of generic vector bundles see [6,7].…”

“…In particular, for generic cochain twists F we have the non-equality 26) for some h ∈ H, L ∈ hom F (V ,W ) and v ∈ V . Using internal homomorphism techniques from category theory, one can show that there exist deformations of the evaluation, composition and tensor product operations which are compatible with the H F -actions [6]. We denote these by…”