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arXiv · hep-th/9410201

Linear Connections in Non-Commutative Geometry

J. Mourad
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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 $Ω^1$. A special role is played by the extension to the framework of non-commutative geometry of the permutation of two copies of $Ω^1$. 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 $Ω^1$. These constructions are illustrated with the example of the algebra of $ n \times n$ matrices.

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J. Mourad. 1994-10-26. Linear Connections in Non-Commutative Geometry. https://doi.org/10.1088/0264-9381%2F12%2F4%2F007

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