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60
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 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 3-dimensional round sphere. The presence of an equivariant real structure J demands a modification in the axiomatic framework of spectral geometry, whereby the commutant and first-order properties need be satisfied only modulo infinitesimals of arbitrary high order.
The spectral action for Moyal planes
- J. Math. Phys
"... Extending a result of D. V. Vassilevich [50], we obtain the asymptotic expansion for the trace of a spatially regularized heat operator LΘ (f)e−t△Θ, where △Θ is a generalized Laplacian defined with Moyal products and LΘ (f) is Moyal left multiplication. The Moyal planes corresponding to any skewsymm ..."
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Cited by 43 (9 self)
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Extending a result of D. V. Vassilevich [50], we obtain the asymptotic expansion for the trace of a spatially regularized heat operator LΘ (f)e−t△Θ, where △Θ is a generalized Laplacian defined with Moyal products and LΘ (f) is Moyal left multiplication. The Moyal planes corresponding to any skewsymmetric matrix Θ being spectral triples [24], the spectral action introduced in noncommutative geometry by A. Chamseddine and A. Connes [6] is computed. This result generalizes the Connes-Lott action [15] previously computed by Gayral [23] for symplectic Θ.
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 Branson-Gilkey 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 well-known 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 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 semi-finite 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 semi-finite 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 q-deformation 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 q-deformation 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.
Some noncommutative geometric aspects of SUq(2)”, math-ph/018003
"... We study various noncommutative geometric aspects of the compact quantum group SUq(2) for positive q (not equal to 1), following the suggestion of Connes and his coauthors ([9], [8]) for considering the so-called true Dirac operator. However, it turns out that the method of the above references do n ..."
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Cited by 13 (5 self)
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We study various noncommutative geometric aspects of the compact quantum group SUq(2) for positive q (not equal to 1), following the suggestion of Connes and his coauthors ([9], [8]) for considering the so-called true Dirac operator. However, it turns out that the method of the above references do not extend to the case of positive (not equal to 1) values of q in the sense that the true Dirac operator does not have bounded commutators with “smooth ” algebra elements in this case, in contrast to what happens for complex q of modulus 1. Nevertheless, we show how to obtain the canonical volume form, i.e. the Haar state, using the true Dirac operator.
Equivariant Poincaré duality for quantum group actions
"... Abstract. We extend the notion of Poincaré duality in KK-theory 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 KK-theory for locally ..."
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Cited by 13 (3 self)
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Abstract. We extend the notion of Poincaré duality in KK-theory 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 KK-theory 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.
