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49
The Dirac operator on SUq(2)
, 2005
"... We construct a 3 +summable spectral triple (A(SUq(2)), H,D) over the quantum group SUq(2) which is equivariant with respect to a left and a right action of Uq(su(2)). The geometry is isospectral to the classical case since the spectrum of the operator D is the same as that of the usual Dirac operat ..."
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Cited by 43 (7 self)
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We construct a 3 +summable spectral triple (A(SUq(2)), H,D) over the quantum group SUq(2) which is equivariant with respect to a left and a right action of Uq(su(2)). The geometry is isospectral to the classical case since the spectrum of the operator D is the same as that of the usual Dirac operator on the 3dimensional round sphere. The presence of an equivariant real structure J demands a modification in the axiomatic framework of spectral geometry, whereby the commutant and firstorder properties need be satisfied only modulo infinitesimals of arbitrary high order.
A local index formula for the quantum sphere
, 2003
"... For the Dirac operator D on the standard quantum sphere we obtain an asymptotic expansion of the SUq(2)equivariant entire cyclic cocycle corresponding to ε 1 2 D when evaluated on the element k2 ∈ Uq(su2). The constant term of this expansion is a twisted cyclic cocycle which up to a scalar coincide ..."
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Cited by 27 (3 self)
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For the Dirac operator D on the standard quantum sphere we obtain an asymptotic expansion of the SUq(2)equivariant entire cyclic cocycle corresponding to ε 1 2 D when evaluated on the element k2 ∈ Uq(su2). The constant term of this expansion is a twisted cyclic cocycle which up to a scalar coincides with the volume form and computes the quantum as well as the classical Fredholm indices.
Spectral Asymmetry, Zeta Functions and the Noncommutative Residue
"... Abstract. In this paper, motivated by an approach developed by Wodzicki, we look at the spectral asymmetry of elliptic ΨDO’s in terms of theirs zeta functions. First, using asymmetry formulas of Wodzicki we study the spectral asymmetry of odd elliptic ΨDO’s and of geometric Dirac operators. In parti ..."
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Cited by 24 (7 self)
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Abstract. In this paper, motivated by an approach developed by Wodzicki, we look at the spectral asymmetry of elliptic ΨDO’s in terms of theirs zeta functions. First, using asymmetry formulas of Wodzicki we study the spectral asymmetry of odd elliptic ΨDO’s and of geometric Dirac operators. In particular, we show that the eta function of a selfadjoint elliptic odd ΨDO is regular at every integer point when the dimension and the order have opposite parities (this generalizes a well known result of BransonGilkey for Dirac operators), and we relate the spectral asymmetry of a Dirac operator on a Clifford bundle to the Riemmanian geometric data, which yields a new spectral interpretation of the Einstein action from gravity. We also obtain a large class of examples of elliptic ΨDO’s for which the regular values at the origin of the (local) zeta functions can easily be seen to be independent of the spectral cut. On the other hand, we simplify the proofs of two wellknown and difficult results of Wodzicki: (i) The independence with respect to the spectral cut of the regular value at the origin of the zeta function of an elliptic ΨDO; (ii) The vanishing of the noncommutative residue of a zero’th order ΨDO projector. These results were proved by Wodzicki using a quite difficult and involved characterization of local invariants of spectral asymmetry, which we can bypass here. Finally, in an appendix we give a new proof of the aforementioned asymmetry formulas of Wodzicki. 1.
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.
THE GARDEN OF QUANTUM SPHERES
, 2002
"... A list of known quantum spheres of dimension one, two and three is presented. ..."
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Cited by 15 (0 self)
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A list of known quantum spheres of dimension one, two and three is presented.
A WALK IN THE NONCOMMUTATIVE GARDEN
"... 2. Handling noncommutative spaces in the wild: basic tools 2 3. Phase spaces of microscopic systems 6 4. Noncommutative quotients 9 ..."
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Cited by 13 (0 self)
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2. Handling noncommutative spaces in the wild: basic tools 2 3. Phase spaces of microscopic systems 6 4. Noncommutative quotients 9
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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