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167
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 32 (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
Quantum symmetry groups of noncommutative spheres
 Commun. Math. Phys
, 2001
"... We show that the noncommutative spheres of Connes and Landi are quantum homogeneous spaces for certain compact quantum groups. We give a general construction of homogeneous spaces which support noncommutative spin geometries. 1 ..."
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Cited by 32 (2 self)
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We show that the noncommutative spheres of Connes and Landi are quantum homogeneous spaces for certain compact quantum groups. We give a general construction of homogeneous spaces which support noncommutative spin geometries. 1
Heatkernel approach to UV/IR mixing on isospectral deformation manifolds
"... We work out the general features of perturbative field theory on noncommutative manifolds defined by isospectral deformation. These (in general curved) ‘quantum spaces’, generalizing Moyal planes and noncommutative tori, are constructed using Rieffel’s theory of deformation quantization by actions o ..."
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Cited by 26 (3 self)
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We work out the general features of perturbative field theory on noncommutative manifolds defined by isospectral deformation. These (in general curved) ‘quantum spaces’, generalizing Moyal planes and noncommutative tori, are constructed using Rieffel’s theory of deformation quantization by actions of R l. Our framework, incorporating background field methods and tools of QFT in curved spaces, allows to deal both with compact and noncompact spaces, as well as with periodic and nonperiodic deformations, essentially in the same way. We compute the quantum effective action up to one loop for a scalar theory, showing the different UV/IR mixing phenomena for different kinds of isospectral deformations. The presence and behavior of the nonplanar parts of the Green functions is understood simply in terms of offdiagonal heat kernel contributions. For periodic deformations, a Diophantine condition on the noncommutivity parameters is found to play a role in the analytical nature of the nonplanar part of the oneloop reduced effective action. Existence of fixed points for the action may give rise to a new kind of UV/IR mixing. Keywords: noncommutative field theory, isospectral deformation, UV/IR mixing, heat kernel, Diophantine approximation.
DBranes, RRFields and Duality ON NONCOMMUTATIVE MANIFOLDS
, 2006
"... We develop some of the ingredients needed for string theory on noncommutative spacetimes, proposing an axiomatic formulation of Tduality as well as establishing a very general formula for Dbrane charges. This formula is closely related to a noncommutative GrothendieckRiemannRoch theorem that is ..."
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Cited by 25 (1 self)
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We develop some of the ingredients needed for string theory on noncommutative spacetimes, proposing an axiomatic formulation of Tduality as well as establishing a very general formula for Dbrane charges. This formula is closely related to a noncommutative GrothendieckRiemannRoch theorem that is proved here. Our approach relies on a very general form of Poincaré duality, which is studied here in detail. Among the technical tools employed are calculations with iterated products in bivariant Ktheory and cyclic theory, which are simplified using a novel diagram calculus reminiscent of Feynman diagrams.
A GROUPOID APPROACH TO QUANTIZATION
, 2007
"... ABSTRACT. Many interesting C ∗algebras can be viewed as quantizations of Poisson manifolds. I propose that a Poisson manifold may be quantized by a twisted polarized convolution C ∗algebra of a symplectic groupoid. Toward this end, I define polarizations for Lie groupoids and sketch the constructi ..."
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Cited by 22 (1 self)
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ABSTRACT. Many interesting C ∗algebras can be viewed as quantizations of Poisson manifolds. I propose that a Poisson manifold may be quantized by a twisted polarized convolution C ∗algebra of a symplectic groupoid. Toward this end, I define polarizations for Lie groupoids and sketch the construction of this algebra. A large number of examples show that this idea unifies previous geometric constructions, including geometric quantization of symplectic manifolds and the C ∗algebra of a Lie groupoid. 1.
Orderunit quantum GromovHausdorff distance
 J. Funct. Anal
, 2003
"... Abstract. We introduce a new distance distoq between compact quantum metric spaces. We show that distoq is Lipschitz equivalent to Rieffel’s distance distq, and give criteria for when a parameterized family of compact quantum metric spaces is continuous with respect to distoq. As applications, we sh ..."
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Cited by 20 (5 self)
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Abstract. We introduce a new distance distoq between compact quantum metric spaces. We show that distoq is Lipschitz equivalent to Rieffel’s distance distq, and give criteria for when a parameterized family of compact quantum metric spaces is continuous with respect to distoq. As applications, we show that the continuity of a parameterized family of quantum metric spaces induced by ergodic actions of a fixed compact group is determined by the multiplicities of the actions, generalizing Rieffel’s work on noncommutative tori and integral coadjoint orbits of semisimple compact connected Lie groups; we also show that the θdeformations of Connes and Landi are continuous in the parameter θ. 1.
QUANTUM ISOMETRIES AND NONCOMMUTATIVE SPHERES
, 905
"... Abstract. We introduce and study two new examples of noncommutative spheres: the halfliberated sphere, and the free sphere. Together with the usual sphere, these two spheres have the property that the corresponding quantum isometry group is “easy”, in the representation theory sense. We present as ..."
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Cited by 20 (10 self)
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Abstract. We introduce and study two new examples of noncommutative spheres: the halfliberated sphere, and the free sphere. Together with the usual sphere, these two spheres have the property that the corresponding quantum isometry group is “easy”, in the representation theory sense. We present as well some general comments on the axiomatization problem, and on the “untwisted ” and “noneasy ” case.
Principal fibrations from noncommutative spheres
, 2004
"... We construct noncommutative principal fibrations S7 θ → S4 θ which are deformations of the classical SU(2) Hopf fibration over the four sphere. We realize the noncommutative vector bundles associated to the irreducible representations of SU(2) as modules of coequivariant maps and construct correspon ..."
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Cited by 19 (8 self)
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We construct noncommutative principal fibrations S7 θ → S4 θ which are deformations of the classical SU(2) Hopf fibration over the four sphere. We realize the noncommutative vector bundles associated to the irreducible representations of SU(2) as modules of coequivariant maps and construct corresponding projections. The index of Dirac operators with coefficients in the associated bundles is computed with the ConnesMoscovici local index formula. The algebra inclusion A(S4 θ) ↩ → A(S7 θ) is an example of a faithfully flat HopfGalois extension (i.e. it can be considered to be a noncommutative principal bundle) which is notcleft (i.e. it is nottrivial).
Dixmier traces on noncompact isospectral deformations
 J. FUNCT. ANAL
, 2005
"... We extend the isospectral deformations of Connes, Landi and DuboisViolette to the case of Riemannian spin manifolds carrying a proper action of the noncompact abelian group R l. Under deformation by a torus action, a standard formula relates Dixmier traces of measurable operators to integrals of fu ..."
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Cited by 18 (8 self)
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We extend the isospectral deformations of Connes, Landi and DuboisViolette to the case of Riemannian spin manifolds carrying a proper action of the noncompact abelian group R l. Under deformation by a torus action, a standard formula relates Dixmier traces of measurable operators to integrals of functions on the manifold. We show that this relation persists for actions of R l, under mild restrictions on the geometry of the manifold which guarantee the Dixmier traceability of those operators.
THE DIRAC OPERATOR ON COMPACT QUANTUM GROUPS
, 2007
"... For the qdeformation Gq, 0 < q < 1, of any simply connected simple compact Lie group G we construct an equivariant spectral triple which is an isospectral deformation of that defined by the Dirac operator D on G. Our quantum Dirac operator Dq is a unitary twist of D considered as an element ..."
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Cited by 18 (8 self)
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For the qdeformation Gq, 0 < q < 1, of any simply connected simple compact Lie group G we construct an equivariant spectral triple which is an isospectral deformation of that defined by the Dirac operator D on G. Our quantum Dirac operator Dq is a unitary twist of D considered as an element of Ug⊗Cl(g). The commutator of Dq with a regular function on Gq consists of two parts. One is a twist of a classical commutator and so is automatically bounded. The second is expressed in terms of the commutator of the associator with an extension of D. We show that in the case of the Drinfeld associator the latter commutator is also bounded.