Modular curvature for toric noncommutative manifolds

Liu, Yang

doi:10.4171/jncg/285

Cited by 23 publications

(51 citation statements)

References 24 publications

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“…The results of this article apply to a class of pseudo-Riemannian bilinear metrics on the space of one forms of a spectral triple (A, H, D) satisfying certain assumptions discussed in this article. Our results do not cover the examples of the conformal perturbations of a Riemannian bilinear metric which is the subject of study of a number of recent works ( [12], [14], [15], [16], [18], [19], [20], [21], [26], [27]). However, in a companion article ( [5]), it is proven that under the same set of assumptions on the spectral triple as in this paper, there exists a unique Levi-Civita connection for any pseudo-Riemannian metric (which is only right A-linear as opposed to being both left and right A-linear.)…”

Section: Introductionmentioning

confidence: 74%

“…Remark 7.15. By Proposition 2.1 of [27], the isotypical decompositions converge absolutely to the element.…”

Section: Some Generalities On Rieffel-deformationmentioning

confidence: 89%

“…Our main reference for Rieffel deformation of a C * algebra endowed with a strongly continuous action by T n is [31]. However, we will also need to use equivalent descriptions of this deformation given in [9], [11], [27]. We next define the deformation of an algebra D. We refer to [27] for details.…”

Section: Some Generalities On Rieffel-deformationmentioning

confidence: 99%

“…Definition 7.14 (Definition 2.3 [27]). Let D be a T n smooth algebra as in Definition 2.2 of [27]. For a skew symmetric n × n matrix θ, consider the bi-character χ θ defined by…”

Section: Some Generalities On Rieffel-deformationmentioning

confidence: 99%

“…In particular, A = C ∞ (M ) and E = Ω 1 D (A). By Sections 2 and 3 of [27], A and E can be deformed to the algebra A θ and the bimodule E θ respectively. Moreover, we have the following: Theorem 7.20.…”

Section: Some Generalities On Rieffel-deformationmentioning

confidence: 99%

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Levi-Civita connections for a class of spectral triples

2019

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We give a new definition of Levi-Civita connection for a noncommutative pseudo-Riemannian metric on a noncommutative manifold given by a spectral triple. We prove the existence-uniqueness result for a class of modules of one forms over a large class of noncommutative manifolds, including the matrix geometry of the fuzzy 3-sphere, the quantum Heisenberg manifolds and Connes-Landi deformations of spectral triples on the Connes-Dubois Violette-Rieffel-deformation of a compact manifold equipped with a free toral action. It is interesting to note that in the example of the quantum Heisenberg manifold, the definition of metric compatibility given in [22] failed to ensure the existence of a unique Levi-Civita connection. In the case of the matrix geometry, the Levi-Civita connection that we get coincides with the unique real torsion-less unitary connection obtained by Frolich et al in [22].

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Section: Introductionmentioning

confidence: 74%

“…Remark 7.15. By Proposition 2.1 of [27], the isotypical decompositions converge absolutely to the element.…”

Section: Some Generalities On Rieffel-deformationmentioning

confidence: 89%

Section: Some Generalities On Rieffel-deformationmentioning

confidence: 99%

“…Definition 7.14 (Definition 2.3 [27]). Let D be a T n smooth algebra as in Definition 2.2 of [27]. For a skew symmetric n × n matrix θ, consider the bi-character χ θ defined by…”

Section: Some Generalities On Rieffel-deformationmentioning

confidence: 99%

Section: Some Generalities On Rieffel-deformationmentioning

confidence: 99%

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Levi-Civita connections for a class of spectral triples

2019

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Curvature in noncommutative geometry

Fathizadeh

Khalkhali

2019

Advances in Noncommutative Geometry

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Dedicated to Alain Connes with admiration, affection, and much appreciation Contents 1. Introduction 2. Curvature in noncommutative geometry 2.1. A brief history of curvature 2.2. Laplace type operators and Gilkey's theorem 2.3. Noncommutative Chern-Weil theory 2.4. From spectral geometry to spectral triples 3. Pseudodifferential calculus and heat expansion 3.1. Classical pseudodifferential calculus 3.2. Small-time heat kernel expansion 3.3. Pseudodifferential calculus and heat kernel expansion for noncommutative tori 4. Gauss-Bonnet theorem and curvature for noncommutative 2-tori 4.1. Scalar curvature and Gauss-Bonnet theorem for T 2 θ 4.2. The Laplacian on (1, 0)-forms on T 2θ with curved metric 5. Noncommutative residues for noncommutative tori and curvature of noncommutative 4-tori 5.1. Noncommutative residues 5.2. Scalar curvature of the noncommutative 4-torus 6.The Riemann curvature tensor and the term a 4 for noncommutative tori 6.1. Functional relations 6.2. A partial differential system associated with the functional relations 6.3. Action of cyclic groups in the differential system, invariant expressions and simple flow of the system 6.4. Gradient calculations leading to functional relations 6.5. The term a 4 for non-conformally flat metrics on noncommutative four tori 7. Twisted spectral triples and Chern-Gauss-Bonnet theorem for ergodic C * -dynamical systems 7.1. Twisted spectral triples 7.2. Conformal perturbation of a spectral triple 7.3. Conformal perturbation of the flat metric on T 2 θ 7.4. Conformally twisted spectral triples for C * -dynamical systems 7.5. The Chern-Gauss-Bonnet theorem for C * -dynamical systems

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Pseudodifferential Operators on Noncommutative Tori: A Survey

Jiménez

2024

La Matematica

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This article presents a survey of recent developments on pseudodifferential operators on noncommutative tori. We describe currently available constructions of those operators: by means of a $$C^{*}$$ C ∗ -dynamical system, by using an analogue of the Fourier series representation of a function in the (commutative) torus $$C^\infty ({\mathbb {T}}^n)$$ C ∞ ( T n ) , as Rieffel deformations of the standard pseudodifferential operators on $$C^\infty ({\mathbb {T}}^n)$$ C ∞ ( T n ) , and in association to certain spectral triples.

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Modular curvature for toric noncommutative manifolds

Cited by 23 publications

References 24 publications

Levi-Civita connections for a class of spectral triples

Levi-Civita connections for a class of spectral triples

Curvature in noncommutative geometry

Pseudodifferential Operators on Noncommutative Tori: A Survey

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