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Lectures on graded differential algebras and noncommutative geometry
, 1999
"... 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. ..."
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Cited by 49 (5 self)
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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.
SU(n)connections and noncommutative differential geometry
 J. Geom. Phys
, 1998
"... We study the noncommutative differential geometry of the algebra of endomorphisms of any SU(n)vector bundle. We show that ordinary connections on such SU(n)vector bundle can be interpreted in a natural way as a noncommutative 1form on this algebra for the differential calculus based on derivation ..."
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Cited by 19 (9 self)
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We study the noncommutative differential geometry of the algebra of endomorphisms of any SU(n)vector bundle. We show that ordinary connections on such SU(n)vector bundle can be interpreted in a natural way as a noncommutative 1form on this algebra for the differential calculus based on derivations. We interpret the Lie algebra of derivations of the algebra of endomorphisms as a Lie algebroid. Then we look at noncommutative connections as generalizations of these usual connections.
Some Aspects of Noncommutative Differential Geometry
"... 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. Finall ..."
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Cited by 18 (1 self)
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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
Submanifolds and Quotient Manifolds in Noncommutative Geometry
 J. Math. Phys
, 1996
"... We define and study noncommutative generalizations of submanifolds and quotient manifolds, for the derivationbased differential calculus introduced by M. DuboisViolette and P. Michor. We give examples to illustrate these definitions. ..."
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Cited by 11 (7 self)
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We define and study noncommutative generalizations of submanifolds and quotient manifolds, for the derivationbased differential calculus introduced by M. DuboisViolette and P. Michor. We give examples to illustrate these definitions.
Noncommutative εgraded connections
, 2012
"... We introduce the new notion of εgraded associative algebras which takes its root into the notion of commutation factors introduced in the context of Lie algebras [1]. We define and study the associated notion of εderivationbased differential calculus, which generalizes the derivationbased differ ..."
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Cited by 10 (2 self)
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We introduce the new notion of εgraded associative algebras which takes its root into the notion of commutation factors introduced in the context of Lie algebras [1]. We define and study the associated notion of εderivationbased differential calculus, which generalizes the derivationbased differential calculus on associative algebras. A corresponding notion of noncommutative connection is also defined. We illustrate these considerations with various examples of εgraded algebras, in particular some graded matrix algebras and the Moyal algebra. This last example permits also to interpret mathematically a noncommutative gauge field theory.
Connections on Central Bimodules
, 1995
"... We define and study the theory of derivationbased connections on a recently introduced class of bimodules over an algebra which reduces to the category of modules whenever the algebra is commutative. This theory contains, in particular, a noncommutative generalization of linear connections. We also ..."
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Cited by 8 (1 self)
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We define and study the theory of derivationbased connections on a recently introduced class of bimodules over an algebra which reduces to the category of modules whenever the algebra is commutative. This theory contains, in particular, a noncommutative generalization of linear connections. We also discuss the different noncommutative versions of differential forms based on derivations. Then we investigate reality conditions and a noncommutative generalization of pseudoriemannian structures.
More on the FrölicherNijenhuis Bracket In Non Commutative Differential Geometry
, 1996
"... . In commutative differential geometry the FrolicherNijenhuis bracket computes all kinds of curvatures and obstructions to integrability. In [1] the FrolicherNijenhuis bracket was developed for universal differential forms of noncommutative algebras, and several applications were given. In this pa ..."
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Cited by 8 (3 self)
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. In commutative differential geometry the FrolicherNijenhuis bracket computes all kinds of curvatures and obstructions to integrability. In [1] the FrolicherNijenhuis bracket was developed for universal differential forms of noncommutative algebras, and several applications were given. In this paper this bracket and the FrolicherNijenhuis calculus will be developed for several kinds of differential graded algebras based on derivations, which were introduced by [6]. Table of contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Convenient vector spaces . . . . . . . . . . . . . . . . . . . . . . 3 3. Preliminaries: graded differential algebras, derivations, and operations of Lie algebras . . . . . . . . . . . . . . . . . . . . 6 4. Derivations on universal differential forms . . . . . . . . . . . . . . . 8 5. The FrolicherNijenhuis calculus on Chevalley type cochains . . . . . . . 11 6. Description of all derivations in the Chevalley differential...
Comments About Higgs Field, Noncommutative Geometry and the Standard Model
 MARSEILLE PREPRINT CPT95 /P.3184 AND HEPTH/9505192
, 1995
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Examples of derivationbased differential calculi related to noncommutative gauge theories
 INTERNATIONAL JOURNAL OF GEOMETRIC METHODS IN MODERN PHYSICS
, 2008
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