Deformation quantization for actions of 𝑅^{𝑑}

Rieffel, Marc A.

doi:10.1090/memo/0506

Cited by 212 publications

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“…where the components c t , c α , c α * , c β , c β * , which are elements in B θ ⊗ B θ ⊗ B θ ⊗ B θ , have an expression of the same form as the corresponding operators in (20)(21)(22)(23)(24) with the symbol d substituted by the tensor product symbol ⊗ . The vanishing of bc, which has six hundred terms, can be checked directly from the commutation relations (9).…”

Section: The Noncommutative 4-spherementioning

confidence: 99%

“…We shall in fact describe a very general construction of isospectral deformations of noncommutative geometries which implies in particular that any compact spin Riemannian manifold M whose isometry group has rank ≥ 2 admits a natural one-parameter isospectral deformation to noncommutative geometries M θ . The deformation of the algebra will be performed along the lines of [23]. We let (A, H, D) be the canonical spectral triple associated with a compact Riemannian spin manifold M. We recall that A = C ∞ (M) is the algebra of smooth functions on M, H = L 2 (M, S) is the Hilbert space of spinors and D is the Dirac operator.…”

Section: Isospectral Deformationsmentioning

confidence: 99%

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Noncommutative Manifolds, the Instanton Algebra¶and Isospectral Deformations

Connes

Landi

2001

Commun. Math. Phys.

289

523

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We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of R n . They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global features.The new examples include the instanton algebra and the NC-4-spheres S 4 θ . We construct the noncommutative algebras A = C ∞ (S 4 θ ) of functions on NCspheres as solutions to the vanishing, ch j (e) = 0 , j < 2, of the Chern character in the cyclic homology of A of an idempotent e ∈ M 4 (A) , e 2 = e , e = e * . We describe the universal noncommutative space obtained from this equation as a noncommutative Grassmanian as well as the corresponding notion of admissible morphisms. This space Gr contains the suspension of a NC-3-sphere intimately related to quantum group deformations SU q (2) of SU(2) but for unusual values (complex values of modulus one) of the parameter q of q-analogues, q = exp(2πiθ).We then construct the noncommutative geometry of S 4 θ as given by a spectral triple (A, H, D) and check all axioms of noncommutative manifolds. In a previous paper it was shown that for any Riemannian metric g µν on S 4 whose volume form √ g d 4 x is the same as the one for the round metric, the corresponding Dirac operator gives a solution to the following quartic equation,where is the projection on the commutant of 4 × 4 matrices. We shall show how to construct the Dirac operator D on the noncommutative 4-spheres S 4 θ so that the previous equation continues to hold without any change. Finally, we show that any compact Riemannian spin manifold whose isometry group has rank r ≥ 2 admits isospectral deformations to noncommutative geometries.

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Section: The Noncommutative 4-spherementioning

confidence: 99%

Section: Isospectral Deformationsmentioning

confidence: 99%

Noncommutative Manifolds, the Instanton Algebra¶and Isospectral Deformations

Connes

Landi

2001

Commun. Math. Phys.

289

523

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“…and each f ∈ C ∞ (M) is the sum of a unique series f = r∈Z n f r , which is rapidly convergent in the Fréchet topology of C ∞ (M) (see [33] for more details). Let now θ = (θ jk = −θ kj ) be a real antisymmetric n × n matrix.…”

Section: Deforming a Torus Actionmentioning

confidence: 99%

“…Indeed, by the very definition of the product × θ in (3.2) one establishes that 8) proving that the algebra C ∞ (M) equipped with the product × θ is isomorphic to the algebra L θ (C ∞ (M)). It is shown in [33] that there is a natural completion of the algebra C ∞ (M θ ) to a C * -algebra C(M θ ) whose smooth subalgebra -under the extended action of T n -is precisely C ∞ (M θ ). Thus, we can understand L θ as a quantization map from…”

Section: Deforming a Torus Actionmentioning

confidence: 99%

Noncommutative Instantons from Twisted Conformal Symmetries

Landi

Suijlekom

2007

Commun. Math. Phys.

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We construct a five-parameter family of gauge-nonequivalent SU (2) instantons on a noncommutative four sphere S 4 θ and of topological charge equal to 1. These instantons are critical points of a gauge functional and satisfy self-duality equations with respect to a Hodge star operator on forms on S 4 θ . They are obtained by acting with a twisted conformal symmetry on a basic instanton canonically associated with a noncommutative instanton bundle on the sphere. A completeness argument for this family is obtained by means of index theorems. The dimension of the "tangent space" to the moduli space is computed as the index of a twisted Dirac operator and turns out to be equal to five, a number that survives deformation.

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“…However, for the purpose of physical applications, we should as well consider quantum groups arising from deformations of kinematical symmetries, as for instance the Heisenberg or the Euclidean groups [13][14][15][16]. This paper deals with the two dimensional Euclidean group, whose q-deformations have been deeply analyzed by many authors [17][18][19][20][21][22][23][24][25]: the novelty of the present approach is that we shall show how different aspects previously considered can be unified by an appropriate use of quantum homogeneous spaces of E q (2), which are recognized as "quantum planes" [26,27] and "quantum hyperboloids" [28]. Since the Euclidean quantum algebra acts canonically on the latter, the action can be used to recover a quantum analog of many results of the classical theory and, in a certain sense, to establish the defining automorphisms of a concrete model of noncommutative geometry.…”

Section: Introductionmentioning

confidence: 99%

Freeq-Schrödinger equation from homogeneous spaces of the 2-dim Euclidean quantum group

et al. 1996

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Abstract. After a preliminary review of the definition and the general properties of the homogeneous spaces of quantum groups, the quantum hyperboloid qH and the quantum plane qP are determined as homogeneous spaces of F q (E(2)). The canonical action of E q (2) is used to define a natural q-analog of the free Schrödinger equation, that is studied in the momentum and angular momentum bases. In the first case the eigenfunctions are factorized in terms of products of two q-exponentials. In the second case we determine the eigenstates of the unitary representation, which, in the qP case, are given in terms of Hahn-Exton functions. Introducing the universal T-matrix for E q (2) we prove that the Hahn-Exton as well as Jackson q-Bessel functions are also obtained as matrix elements of T, thus giving the correct extension to quantum groups of well known methods in harmonic analysis.Mathematics Subject Classification: 16W30, 17B37, 22E99.

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Deformation quantization for actions of 𝑅^{𝑑}

Cited by 212 publications

References 0 publications

Noncommutative Manifolds, the Instanton Algebra¶and Isospectral Deformations

Noncommutative Manifolds, the Instanton Algebra¶and Isospectral Deformations

Noncommutative Instantons from Twisted Conformal Symmetries

Freeq-Schrödinger equation from homogeneous spaces of the 2-dim Euclidean quantum group

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