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35
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.
Hamiltonian Gravity and Noncommutative Geometry
"... A version of foliated spacetime is constructed in which the spatial geometry is described as a time dependent noncommutative geometry. The ADM version of the gravitational action is expressed in terms of these variables. It is shown that the vector constraint is obtained without the need for an e ..."
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Cited by 14 (1 self)
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A version of foliated spacetime is constructed in which the spatial geometry is described as a time dependent noncommutative geometry. The ADM version of the gravitational action is expressed in terms of these variables. It is shown that the vector constraint is obtained without the need for an extraneous shift vector in the action. Introduction The problem of divergences in quantum field theory, and especially in quantum gravity, strongly suggests a need to describe the geometry of spacetime as something different or more general than a manifold of points. One interesting generalization of classical geometry is noncommutative differential geometry. It has attracted some attention in physics in recent years, mostly in the form of ConnesLott and related models of particle physics [3]. It has also been used as a regularization technique for Euclidean quantum field theory [6]. Attention to applying noncommutative geometry to gravitation has been limited thus far; an expression for...
Noncommutative and SemiRiemannian Geometry” J.Geom.Phys
, 2006
"... We introduce the notion of a semiRiemannian spectral triple which generalizes the notion of spectral triple and allows for a treatment of semiRiemannian manifolds within a noncommutative setting. It turns out that the relevant spaces in noncommutative semiRiemannian geometry are not Hilbert space ..."
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Cited by 14 (0 self)
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We introduce the notion of a semiRiemannian spectral triple which generalizes the notion of spectral triple and allows for a treatment of semiRiemannian manifolds within a noncommutative setting. It turns out that the relevant spaces in noncommutative semiRiemannian geometry are not Hilbert spaces any more but Krein spaces, and Dirac operators are Kreinselfadjoint. We show that the noncommutative tori can be endowed with a semiRiemannian structure in this way. For the noncommutative tori as well as for semiRiemannian spin manifolds the dimension, the signature of the metric, and the integral of a function can be recovered from the spectral data.
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
Sukochev The Chern Character of Semifinite Spectral Triples
"... All authors were supported by grants from ARC (Australia) and NSERC (Canada), in addition ..."
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Cited by 9 (6 self)
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All authors were supported by grants from ARC (Australia) and NSERC (Canada), in addition
THE DIXMIER TRACE AND ASYMPTOTICS OF ZETA FUNCTIONS
, 2006
"... We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the trace of the heat semigroup. We prove our results in a general semifinite von Neumann algebra. We find for p> 1 that the asymptotics of the zeta function determ ..."
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Cited by 6 (3 self)
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We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the trace of the heat semigroup. We prove our results in a general semifinite von Neumann algebra. We find for p> 1 that the asymptotics of the zeta function determines an ideal strictly larger than L p, ∞ on which the Dixmier trace may be defined. We also establish stronger versions of other results on Dixmier traces and zeta functions.