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.

A review is made of recent efforts to add a gravitational field to noncommutative models of space-time. Special emphasis is placed on the case which could be considered as the noncommutative analog of a parallelizable space-time. It is argued that, at least in this case, there is a rigid relation between the noncommutative structure of the space-time on the one hand and the nature of the gravitational field which remains as a `shadow' in the commutative limit on the other. ESI Preprint 478 (1997). Lecture given at the International Workshop "Mathematical Physics - today, Priority Technologies - for tomorrow", Kyiv, Ukraine, May 1997. Laboratoire associ'e au CNRS, URA D0063 1 Introduction and Motivation Simply stated, `fuzzy space-time' is a space-time in which the `coordinates' do not commute. One typically replaces the four Minkowski coordinates x ¯ by four generators q ¯ of a noncommutative algebra which satisfy commutation relations of the form [q ¯ ; q ] = i¯kq ¯ :...

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.

Abstract. In this article we review some results obtained from a generalization of quantum mechanics obtained from modification of the canonical commutation relation [q, p] = i¯h. We present some new results concerning relativistic generalizations of previous works, and we calculate the energy spectrum of some simple quantum systems, using the position and momentum operators of this new formalism.

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A revision of generalized commutation relations is performed, besides a description of Non linear momenta realization included in some DSR theories. It is shown that these propositions are closely related, specially we focus on Magueijo Smolin momenta and Kempf et al. and L.N. Chang generalized commutators. Due to this, a new algebra arises with its own features that is also analyzed.

A revision of generalized commutation relations is performed, beside a description of Deformed Special Relativity (DSR) theories. It is demonstrated that these propositions are very closely related, specially Magueijo Smolin momenta and Kempf et al. and L.N. Chang generalized commutators. Due this, a new algebra arise with its own features that is also analyzed. 1