Nonassociative geometry in quasi-Hopf representation categories II: Connections and curvature

Abstract

We continue our systematic development of noncommutative and nonassociative differential geometry internal to the representation category of a quasitriangular quasi-Hopf algebra. We describe derivations, differential operators, differential calculi and connections using universal categorical constructions to capture algebraic properties such as Leibniz rules. Our main result is the construction of morphisms which provide prescriptions for lifting connections to tensor products and to internal homomorphisms. We describe the curvatures of connections within our formalism, and also the formulation of Einstein-Cartan geometry as a putative framework for a nonassociative theory of gravity.

29 pages. v2: Final version published in Journal of Geometry and Physics

References in corpus (6)

• T-duality and closed string non-commutative (doubled) geometry
• Noncommutative D=4 gravity coupled to fermions
• Nonassociative geometry in quasi-Hopf representation categories I: Bimodules and their internal homomorphisms
• Twisted Poisson Structures and Non-commutative/non-associative Closed String Geometry
• Triproducts, nonassociative star products and geometry of R-flux string compactifications
• Twisting all the way: from algebras to morphisms and connections

Cited by in corpus (19)

• Non-geometric backgrounds in string theory
• $G_2$-structures and quantization of non-geometric M-theory backgrounds
• Nonassociative differential geometry and gravity with non-geometric fluxes
• Braided $L_{\infty }$-Algebras, Braided Field Theory and Noncommutative Gravity
• Towards a Theory of Nonassociative Gravity
• Braided Cartan Calculi and Submanifold Algebras
• Braided Symmetries in Noncommutative Field Theory
• Noncommutative principal bundles through twist deformation
• $L_{\infty }$-Algebras of Einstein-Cartan-Palatini Gravity
• Magnetic monopoles and nonassociative deformations of quantum theory
• Global Double Field Theory is Higher Kaluza-Klein Theory
• Weak associativity and deformation quantization
• Cartan structure equations and Levi-Civita connection in noncommutative geometry
• Nonassociative geometry of nonholonomic phase spaces with star R-flux string deformations and (non) symmetric metrics
• Working with Nonassociative Geometry and Field Theory
• Homotopy Lie Algebras of Gravity and their Braided Deformations
• On the Relationship between Classical and Deformed Hopf Fibrations
• Topics in Noncommutative Geometry
• Extended Field Theories as higher Kaluza-Klein theories