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12
Noncommutative FiniteDimensional Manifolds  I. SPHERICAL MANIFOLDS AND RELATED EXAMPLES
, 2001
"... We exhibit large classes of examples of noncommutative finitedimensional manifolds which are (nonformal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative threedimensional spherical manifolds, a noncommutative version of the sphere S 3 d ..."
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Cited by 90 (12 self)
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We exhibit large classes of examples of noncommutative finitedimensional manifolds which are (nonformal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative threedimensional spherical manifolds, a noncommutative version of the sphere S 3 defined by basic Ktheoretic equations. We find a 3parameter family of deformations of the standard 3sphere S 3 and a corresponding 3parameter deformation of the 4dimensional Euclidean space R 4. For generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed R 4 u are isomorphic to the algebras introduced by Sklyanin in connection with the YangBaxter equation. Special values of the deformation parameters do not give rise to Sklyanin algebras and we extract a subclass, the θdeformations, which we generalize in any dimension and various contexts, and study in some details. Here, and
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 22 (3 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.
The Fuzzy Supersphere
, 1998
"... We introduce the fuzzy supersphere as sequence of finitedimensional, noncommutative Z2graded algebras tending in a suitable limit to a dense subalgebra of the Z2graded algebra of H1functions on the (22)dimensional supersphere. Noncommutative analogues of the body map (to the (fuzzy) sphere) ..."
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Cited by 17 (2 self)
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We introduce the fuzzy supersphere as sequence of finitedimensional, noncommutative Z2graded algebras tending in a suitable limit to a dense subalgebra of the Z2graded algebra of H1functions on the (22)dimensional supersphere. Noncommutative analogues of the body map (to the (fuzzy) sphere) and the superdeRham complex are introduced. In particular we reproduce the equality of the superdeRham cohomology of the supersphere and the ordinary deRham cohomology of its body on the "fuzzy level".
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 12 (2 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
On the noncommutative geometry of the endomorphism algebra of a vector bundle
 Masson T., Submanifolds and quotient manifolds in noncommutative
, 1999
"... In this letter we investigate some aspects of the noncommutative differential geometry based on derivations of the algebra of endomorphisms of an oriented complex hermitian vector bundle. We relate it, in a natural way, to the geometry of the underlying principal bundle and compute the cohomology of ..."
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Cited by 12 (4 self)
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In this letter we investigate some aspects of the noncommutative differential geometry based on derivations of the algebra of endomorphisms of an oriented complex hermitian vector bundle. We relate it, in a natural way, to the geometry of the underlying principal bundle and compute the cohomology of its complex of noncommutative differential forms.
Trout: Representable ETheory for C0(X)algebras
 Journal of Functional Analysis
, 2000
"... Abstract. Let X be a locally compact space, and let A and B be C0(X)algebras. We define the notion of an asymptotic C0(X)morphism from A to B and construct representable Etheory groups RE(X; A, B). These are the universal groups on the category of separable C0(X)algebras that are C0(X)stable, C ..."
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Cited by 5 (0 self)
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Abstract. Let X be a locally compact space, and let A and B be C0(X)algebras. We define the notion of an asymptotic C0(X)morphism from A to B and construct representable Etheory groups RE(X; A, B). These are the universal groups on the category of separable C0(X)algebras that are C0(X)stable, C0(X)homotopyinvariant, and halfexact. If A is RKK(X)nuclear, these groups are naturally isomorphic to Kasparov’s representable KKtheory groups RKK(X; A, B). Applications and examples are also discussed. 1.
Examples of derivationbased differential calculi related to noncommutative gauge theories
 INTERNATIONAL JOURNAL OF GEOMETRIC METHODS IN MODERN PHYSICS
, 2008
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Derivations of the Moyal Algebra and Noncommutative Gauge Theories
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2009
"... The differential calculus based on the derivations of an associative algebra underlies most of the noncommutative field theories considered so far. We review the essential properties of this framework and the main features of noncommutative connections in the case of non graded associative unital a ..."
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Cited by 2 (1 self)
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The differential calculus based on the derivations of an associative algebra underlies most of the noncommutative field theories considered so far. We review the essential properties of this framework and the main features of noncommutative connections in the case of non graded associative unital algebras with involution. We extend this framework to the case of Z2graded unital involutive algebras. We show, in the case of the Moyal algebra or some related Z2graded version of it, that the derivation based differential calculus is a suitable framework to construct Yang–Mills–Higgs type models on Moyal (or related) algebras, the covariant coordinates having in particular a natural interpretation as Higgs fields. We also exhibit, in one situation, a link between the renormalisable NC ϕ4model with harmonic term and a gauge theory model. Some possible consequences of this are briefly discussed.
Noncommutative generalization of SU(n)principal ber bundles: a review
, 709
"... Abstract. This is an extended version of a communication made at the international conference Noncommutative Geometry and Physics held at Orsay in april 2007. In this proceeding, we make a review of some noncommutative constructions connected to the ordinary ber bundle theory. The noncommutative alg ..."
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Abstract. This is an extended version of a communication made at the international conference Noncommutative Geometry and Physics held at Orsay in april 2007. In this proceeding, we make a review of some noncommutative constructions connected to the ordinary ber bundle theory. The noncommutative algebra is the endomorphism algebra of a SU(n)vector bundle, and its di erential calculus is based on its Lie algebra of derivations. It is shown that this noncommutative geometry contains some of the most important constructions introduced and used in the theory of connections on vector bundles, in particular, what is needed to introduce gauge models in physics, and it also contains naturally the essential aspects of the Higgs elds and its associated mechanics of mass generation. It permits one also to extend some previous constructions, as for instance symmetric reduction of (here noncommutative) connections. From a mathematical point of view, these geometricoalgebraic considerations highlight some new point on view, in particular we introduce a new construction of the Chern characteristic classes. LPTOrsay/0757