Differential Calculi and Linear Connections

Abstract

A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found that there is a strong correlation, but not a one-to-one correspondence, between the module structure of the 1-forms and the metric torsion-free connections on it. In the commutative limit the connection remains as a shadow of the algebraic structure of the 1-forms.

16 pages of LaTeX

References in corpus (1)

• Non-commutative Geometry and Kinetic Theory of Open Systems

Cited by in corpus (19)

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• Topology at the Planck Length
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• Field Theory on Curved Noncommutative Spacetimes
• The Geometry of the Quantum Euclidean Space
• Non Commutative Differential Geometry, and the Matrix Representations of Generalised Algebras
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• Classical Gravity on Fuzzy Space-Time
• Fuzzy Surfaces of Genus Zero
• Automorphisms of associative algebras and noncommutative geometry
• The Hidden Geometry of the Quantum Euclidean Space
• Frame formalism for the N-dimensional quantum Euclidean spaces