Connections on central bimodules

Abstract

We define and study the theory of derivation-based connections on a recently introduced class of bimodules over an algebra which reduces to the category of modules whenever the algebra is commutative. This theory contains, in particular, a noncommutative generalization of linear connections. We also discuss the different noncommutative versions of differential forms based on derivations. Then we investigate reality conditions and a noncommutative generalization of pseudo-riemannian structures.

27 pages, AMS-LaTeX

References in corpus (4)

• Linear Connections in Non-Commutative Geometry
• Linear connections on the quantum plane
• Linear connections on matrix geometries
• The Frölicher-Nijenhuis bracket for derivation based non commutative differential forms

Cited by in corpus (68)

• An Introduction to Noncommutative Spaces and their Geometry
• On Curvature in Noncommutative Geometry
• Noncommutative Geometry of Finite Groups
• Quantum Space-time and Classical Gravity
• Linear connections on the quantum plane
• Gravity induced from quantum spacetime
• Linear connections on matrix geometries
• Noncommutative epsilon-graded connections
• The Fuzzy Supersphere
• Noncommutative geometry, gauge theory and renormalization
• Riemannian Geometry of Noncommutative Surfaces
• Gauge theories on quantum spaces
• Connections on Lie algebroids and on derivation-based noncommutative geometry
• Quantum gravity on a square graph
• Submanifolds and Quotient Manifolds in Noncommutative Geometry
• Quantum Riemannian geometry and particle creation on the integer line
• Almost commutative Riemannian geometry: wave operators
• Exact Partition Functions for Gauge Theories on ${\mathbb{R}}_{\lambda }^3$
• Braided Cartan Calculi and Submanifold Algebras
• Noncommutative Geometry of Super-Jordanian $OS{p}_h(2/1)$ Covariant Quantum Space
• On the Koszul formula in noncommutative geometry
• Linear Connections on the Two Parameter Quantum Plane
• Quantum geodesic flows and curvature
• Noncommutative spherically symmetric spacetimes at semiclassical order
• On the Noncommutative Geometry of the Endomorphism Algebra of a Vector Bundle
• Quantum geodesics on $\lambda$-Minkowski spacetime
• Quantum gravity on polygons and $\mathbb{R}\times {\mathbb{Z}}_n$ FLRW model
• Levi-Civita connections and vector fields for noncommutative differential calculi
• Fuzzy and discrete black hole models
• Classification of digital affine noncommutative geometries
• Emergence of Riemannian geometry and the massive graviton
• Examples of derivation-based differential calculi related to noncommutative gauge theories
• Derivations of the Moyal Algebra and Noncommutative Gauge Theories
• Digital finite quantum Riemannian geometries
• Cartan structure equations and Levi-Civita connection in noncommutative geometry
• Lectures on Differential Geometry of Modules and Rings
• Geometric Dirac operator on the fuzzy sphere
• On the emergence of the structure of Physics
• Quantum Koszul formula on quantum spacetime
• Hopf Algebroids, Bimodule Connections and Noncommutative Geometry
• Noncommutative generalization of SU(n)-principal fiber bundles: a review
• Twisted submanifolds of R^n
• Positive Line Bundles Over the Irreducible Quantum Flag Manifolds
• Quantum Kaluza-Klein theory with $M_2(\mathbb{C})$
• Quantum Gravity and Riemannian Geometry on the Fuzzy Sphere
• Linear Connections on Graphs
• Quantum Riemannian geometry of the discrete interval and q-deformation
• Bimodule Connections for Relative Line Modules over the Irreducible Quantum Flag Manifolds
• Yang-Mills field from fuzzy sphere quantum Kaluza-Klein model
• On The non-commutative Riemannian geometry of GL_q(n)
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• Module parallel transports in fuzzy gauge theory
• Induced $C^{\ast }$-complexes in metaplectic geometry
• Empirical topology in the histories approach to quantum theory
• Finite noncommutative geometries related to $F_p[x]$
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• Axiomatic classical (prequantum) field theory. Jet formalism
• Levi-Civita Connections on the Quantum Groups SL_{q}(N), O_{q}(N) and Sp_{q}(N)
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• Derivation-based Noncommutative Field Theories on $AF$ algebras
• Projective real calculi over matrix algebras
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• Quantum Calculi: differential forms and vector fields in noncommutative geometry
• Poisson Principal Bundles
• Foundations of Noncommutative Carrollian Geometry via Lie-Rinehart Pairs
• Isometric Spectral Subtriples