These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.

L.P.T.H.E.-ORSAY 95/78 ESI-preprint 285 1Laboratoire associ'e au Centre National de la Recherche Scientifique- URA D0063 1 Abstract We discuss in some generality aspects of noncommutative differential geometry associated with reality conditions and with differential calculi. We then describe the differential calculus based on derivations as generalization of vector fields, and we show its relations with quantum mechanics. Finally we formulate a general theory of connections in this framework. 1 Introduction In [23], J.L. Koszul described a powerful algebraic version of differential geometry in terms of a commutative associative algebra C, C-modules and connections ("derivation laws") on these modules. For the applications to differential geometry, C is the algebra of smooth functions on a manifold and the C-modules are modules of smooth sections of smooth vector bundles over the manifold. The fact that classical differential geometry admits such an algebraic formulation is at the very origin of the idea of noncommutative differential geometry. Historically, the motivation of noncommutative geometry was the development of quantum theory [12]. In noncommutative geometry, one replaces the commutative associative algebra C by an associative algebra A which is not assumed to be commutative. However this replacement raises several problems which will be discussed in this lecture.

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We analyse the structure of the κ = 0 limit of a family of algebras Aκ describing noncommutative versions of space-time, with κ a parameter of noncommutativity. Assuming the Poincaré covariance of the κ = 0 limit, we show that, besides the algebra of functions on Minkowski space, A0 must contain a nontrivial extra factor AI 0 which is Lorentz covariant and which does not commute with the functions whenever it is not commutative. We give a general description of the possibilities and analyse some representative examples. 1

We analyse the structure of the κ = 0 limit of a family of algebras Aκ describing noncommutative versions of space-time, with κ a parameter of noncommutativity. Assuming the Poincaré covariance of the κ = 0 limit, we show that, besides the algebra of functions on Minkowski space, A0 must contain a nontrivial extra factor AI 0 which is Lorentz covariant and which does not commute with the functions whenever it is not commutative. We give a general description of the possibilities and analyse some representative examples. 1