Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1994-10-26
Class.Quant.Grav.12:965-974,1995
Physics
High Energy Physics
High Energy Physics - Theory
15 pages, LMPM ../94 (uses phyzzx)
Scientific paper
10.1088/0264-9381/12/4/007
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.
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