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43
Quantum group of isometries in classical and noncommutative geometry
 Comm. Math. Phys
"... We formulate a quantum generalization of the notion of the group of Riemannian isometries for a compact Riemannian manifold, by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or noncommutative manifold described by spectral triples, and then pro ..."
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Cited by 45 (21 self)
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We formulate a quantum generalization of the notion of the group of Riemannian isometries for a compact Riemannian manifold, by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or noncommutative manifold described by spectral triples, and then proving the existence of a universal object (called the quantum isometry group) in the category of compact quantum groups acting smoothly and isometrically on a given (possibly noncommutative) manifold. Our formulation accommodates spectral triples which are not of type II. We give explicit description of quantum isometry groups of commutative and noncommutative tori, and in this context, obtain the quantum double torus defined in [7] as the universal quantum group of holomorphic isometries of the noncommutative torus. 1
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 det ..."
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Cited by 21 (11 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.
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.
Dirac operators on all Podles quantum spheres
, 2006
"... We construct spectral triples on all Podles quantum spheres S²qt. These noncommutative geometries are equivariant for a left action of Uq(su(2)) and are regular, even and of metric dimension 2. They are all isospectral to the undeformed round geometry of the sphere S². There is also an equivariant r ..."
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Cited by 14 (1 self)
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We construct spectral triples on all Podles quantum spheres S²qt. These noncommutative geometries are equivariant for a left action of Uq(su(2)) and are regular, even and of metric dimension 2. They are all isospectral to the undeformed round geometry of the sphere S². There is also an equivariant real structure for which both the commutant property and the first order condition for the Dirac operators are valid up to infinitesimals of arbitrary order.
Equivariant Poincaré duality for quantum group actions
"... Abstract. We extend the notion of Poincaré duality in KKtheory to the setting of quantum group actions. An important ingredient in our approach is the replacement of ordinary tensor products by braided tensor products. Along the way we discuss general properties of equivariant KKtheory for locally ..."
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Cited by 13 (3 self)
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Abstract. We extend the notion of Poincaré duality in KKtheory to the setting of quantum group actions. An important ingredient in our approach is the replacement of ordinary tensor products by braided tensor products. Along the way we discuss general properties of equivariant KKtheory for locally compact quantum groups, including the construction of exterior products. As an example, we prove that the standard Podle´s sphere is equivariantly Poincaré dual to itself. 1.
On the noncommutative spin geometry of the standard Podle´s sphere and index computations
, 2008
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Quantum isometry groups of the Podles sphere
"... For µ ∈ [0, 1], c ≥ 0, We identify the quantum group SOµ(3) as the universal object in the category of compact quantum groups acting ‘by orientation and volume preserving isometries ’ in the sense of [8] on the natural spectral triple on the Podles sphere S 2 µ,c constructed by Dabrowski, D’Andrea, ..."
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Cited by 10 (6 self)
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For µ ∈ [0, 1], c ≥ 0, We identify the quantum group SOµ(3) as the universal object in the category of compact quantum groups acting ‘by orientation and volume preserving isometries ’ in the sense of [8] on the natural spectral triple on the Podles sphere S 2 µ,c constructed by Dabrowski, D’Andrea, Landi and Wagner in [12]. Moreover, we explicitly compute such universal quantum groups for another class of spectral triples on S2 µ,c ( c> 0) constructed by Chakraborty and Pal ([9]). 1
A residue formula for the fundamental Hochschild class of the Podleś sphere
 Journal of Ktheory: Ktheory and its Applications to Algebra, Geometry, and Topology
, 2013
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Quantum group of orientationpreserving Riemannian isometries
 J. Funct. Anal
"... We formulate a quantum group analogue of the group of orientationpreserving Riemannian isometries of a compact Riemannian spin manifold, more generally, of a (possibly Rtwisted and of compact type) spectral triple. The main advantage of this formulation, which is directly in terms of the Dirac oper ..."
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Cited by 8 (2 self)
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We formulate a quantum group analogue of the group of orientationpreserving Riemannian isometries of a compact Riemannian spin manifold, more generally, of a (possibly Rtwisted and of compact type) spectral triple. The main advantage of this formulation, which is directly in terms of the Dirac operator, is that it does not need the existence of any ‘good ’ Laplacian as in our previous works on quantum isometry groups. Several interesting examples, including those coming from Rieffeltype deformation as well as the equivariant spectral triples on SUµ(2) and S2 µc are dicussed.
Quantum isometries of the finite noncommutative geometry of the standard model
 Comm. Math. Phys
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