Linear Connections in Non-Commutative Geometry

Abstract

A construction is proposed for linear connections on non-commutative algebras. The construction relies on a generalisation of the Leibnitz rules of commutative geometry and uses the bimodule structure of ${\Omega }^1$. A special role is played by the extension to the framework of non-commutative geometry of the permutation of two copies of ${\Omega }^1$. The construction of the linear connection as well as the definition of torsion and curvature is first proposed in the setting of the derivations based differential calculus of Dubois- Violette and then a generalisation to the framework proposed by Connes as well as other non-commutative differential calculi is suggested. The covariant derivative obtained admits an extension to the tensor product of several copies of ${\Omega }^1$. These constructions are illustrated with the example of the algebra of $n\times n$ matrices.

15 pages, LMPM ../94 (uses phyzzx)

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Cited by in corpus (70)

• An Introduction to Noncommutative Spaces and their Geometry
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• Differential Calculi and Linear Connections
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• Gauge theories on quantum spaces
• Nonassociative geometry in quasi-Hopf representation categories II: Connections and curvature
• Quantum gravity on a square graph
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• Noncommutative Geometry for Pedestrians
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• Noncommutative principal bundles through twist deformation
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• Classification of digital affine noncommutative geometries
• LINEAR CONNECTIONS ON EXTENDED SPACE-TIME
• Examples of derivation-based differential calculi related to noncommutative gauge theories
• Lectures on graded differential algebras and noncommutative geometry
• Riemannian Geometry of a Discretized Circle and Torus
• Digital finite quantum Riemannian geometries
• Lectures on Differential Geometry of Modules and Rings
• Twisting all the way: from algebras to morphisms and connections
• ON THE ORIGIN OF KALUZA-KLEIN STRUCTURE
• Noncommutative 2-Dimensional Models of Gravity
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• Classical Gravity on Fuzzy Space-Time
• Linear Connections on Graphs
• Yang-Mills field from fuzzy sphere quantum Kaluza-Klein model
• Noncommutative geometry of phase space
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• Deformation quantization of principal bundles
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