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Linear connections in non-commutative geometry. (English) Zbl 0822.58006

Summary: A construction is proposed for linear connection on non-commutative algebras. The construction relies on a generalization of the Leibniz 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 generalization to the framework of the Dirac operator based differential calculus of Connes and other differential calculuses is given. 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.

MSC:

46L85 Noncommutative topology
46L87 Noncommutative differential geometry
83C47 Methods of quantum field theory in general relativity and gravitational theory
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