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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 89 (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
Strong connections on quantum principal bundles
 Commun. Math. Phys
, 1996
"... A gauge invariant notion of a strong connection is presented and characterized. It is then used to justify the way in which a global curvature form is defined. Strong connections are interpreted as those that are induced from the base space of a quantum bundle. Examples of both strong and nonstrong ..."
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Cited by 37 (7 self)
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A gauge invariant notion of a strong connection is presented and characterized. It is then used to justify the way in which a global curvature form is defined. Strong connections are interpreted as those that are induced from the base space of a quantum bundle. Examples of both strong and nonstrong connections are provided. In particular, such connections are constructed on a quantum deformation of the Hopf fibration S 2 → RP 2. A certain class of strong Uq(2)connections on a trivial quantum principal bundle is shown to be equivalent to the class of connections on a free module that are compatible with the qdependent hermitian metric. A particular form of the Yang–Mills action on a trivial Uq(2)bundle is investigated. It is proved to coincide with the Yang–Mills action constructed by A. Connes and M. Rieffel. Furthermore, it is shown that the moduli space of critical points of this action functional is independent of q.
MODULI SPACE AND STRUCTURE OF NONCOMMUTATIVE 3SPHERES
, 2003
"... We analyse the moduli space and the structure of noncommutative 3spheres. We develop the notion of central quadratic form for quadratic algebras, and prove a general algebraic result which considerably refines the classical homomorphism from a quadratic algebra to a crossproduct algebra associated ..."
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Cited by 26 (9 self)
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We analyse the moduli space and the structure of noncommutative 3spheres. We develop the notion of central quadratic form for quadratic algebras, and prove a general algebraic result which considerably refines the classical homomorphism from a quadratic algebra to a crossproduct algebra associated to the characteristic variety and lands in a richer crossproduct. It allows to control the C ∗norm on involutive quadratic algebras and to construct the differential calculus in the desired generality. The moduli space of noncommutative 3spheres is identified with equivalence classes of pairs of points in a symmetric space of unitary unimodular symmetric matrices. The scaling foliation of the moduli space is identified to the gradient flow of the character of a virtual representation of SO(6). Its generic orbits are connected components of real parts of elliptic curves which form a net of biquadratic curves with 8 points in common. We show
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.
Differential calculi and linear connections
 J. Math. Phys
, 1996
"... A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found that there is a strong correlation, but not a onetoone correspondence, between the module structure of ..."
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Cited by 21 (13 self)
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A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found that there is a strong correlation, but not a onetoone correspondence, between the module structure of the 1forms and the metric torsionfree connections on it. In the commutative limit the connection remains as a shadow of the algebraic structure of the 1forms.
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 18 (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".
Gravity on fuzzy spacetime
, 1992
"... Dedicated to Walter Thirring on the occasion of his 70th birthday A review is made of recent efforts to add a gravitational field to noncommutative models of spacetime. Special emphasis is placed on the case which could be considered as the noncommutative analog of a parallelizable spacetime. It i ..."
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Cited by 15 (0 self)
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Dedicated to Walter Thirring on the occasion of his 70th birthday A review is made of recent efforts to add a gravitational field to noncommutative models of spacetime. Special emphasis is placed on the case which could be considered as the noncommutative analog of a parallelizable spacetime. It is argued that, at least in this case, there is a rigid relation between the noncommutative structure of the spacetime 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
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 13 (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.
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 13 (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