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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 126 (15 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 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.
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 25 (3 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".
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 19 (9 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.
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
2010, Almost commutative Riemannian geometry, I: wave operators
"... Abstract. Associated to any (pseudo)Riemannian manifold M of dimension n is an n + 1dimensional noncommutative differential structure (Ω1, d) on the manifold, with the extra dimension encoding the classical Laplacian as a noncommutative ‘vector field’. We use the classical connection, Ricci tenso ..."
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Cited by 12 (7 self)
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Abstract. Associated to any (pseudo)Riemannian manifold M of dimension n is an n + 1dimensional noncommutative differential structure (Ω1, d) on the manifold, with the extra dimension encoding the classical Laplacian as a noncommutative ‘vector field’. We use the classical connection, Ricci tensor and Hodge Laplacian to construct (Ω2, d) and a natural noncommutative torsion free connection (∇, σ) on Ω1. We show that its generalised braiding σ: Ω1 ⊗ Ω1 → Ω1 ⊗ Ω1 obeys the quantum YangBaxter or braid relations only when the original M is flat, i.e their failure is governed by the Riemann curvature, and that σ2 = id only when M is Einstein. We show that if M has a conformal Killing vector field τ then the cross product algebra C(M) oτ R viewed as a noncommutative analogue of M × R has a natural n + 2dimensional calculus extending Ω1 and a natural spacetime Laplacian now directly defined by the extra dimension. The case M = R3 recovers the MajidRuegg bicrossproduct flat spacetime model and the waveoperator used in its variable speed of light preduction, but now as an example of a general construction. As an application we construct the wave operator on a noncommutative Schwarzschild black hole and take a first look at its features. It appears that the infinite classical redshift/time dilation factor at the event horizon is made finite. 1.
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 12 (3 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.
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 7 (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.
Noncommutative Riemannian geometry of graphs
 J. Geom. Phys
, 2013
"... Abstract. We show that arising out of noncmmutatve geometry is a natural family of edge Laplacians on the edges of a graph. The family includes a canonical edge Laplacian associated to the graph, extending the usual graph Laplacian on vertices, and we find its spectrum. We show that for a connecte ..."
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Cited by 6 (3 self)
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Abstract. We show that arising out of noncmmutatve geometry is a natural family of edge Laplacians on the edges of a graph. The family includes a canonical edge Laplacian associated to the graph, extending the usual graph Laplacian on vertices, and we find its spectrum. We show that for a connected graph its eigenvalues are strictly positive aside from one mandatory zero mode, and include all the vertex degrees. Our edge Laplacian is not the graph Laplacian on the line graph but rather it arises as the noncommutative LaplaceBeltrami operator on differential 1forms, where we use the language of differential algebras to functorially interpret a graph as providing a ‘finite manifold structure ’ on the set of vertices. We equip any graph with a canonical ‘Euclidean metric ’ and a canonical bimodule connection, and in the case of a Cayley graph we construct a metric compatible connection for the Euclidean metric. We make use of results on bimodule connections on inner calculi on algebras, which we prove, including a general relation between zero curvature and the braid relations. 1.