Principal fibrations from noncommutative spheres

Abstract

We construct noncommutative principal fibrations S_θ^7 \to S_θ^4 which are deformations of the classical SU(2) Hopf fibration over the four sphere. We realize the noncommutative vector bundles associated to the irreducible representations of SU(2) as modules of coequivariant maps and construct corresponding projections. The index of Dirac operators with coefficients in the associated bundles is computed with the Connes-Moscovici local index formula. The algebra inclusion $A(S_θ^4) \into A(S_θ^7)$ is an example of a not trivial quantum principal bundle.

23 pages. Latex. v3: Additional minor corrections, version published in CMP

References in corpus (3)

• Bijectivity of the canonical map for the noncommutative instanton bundle
• On the Noncommutative Geometry of Twisted Spheres
• Hopf fibration and monopole connection over the contact quantum spheres

Cited by in corpus (27)

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• Noncommutative instantons from twisted conformal symmetries
• Gauge theory on noncommutative Riemannian principal bundles
• Analytic and algebraic indices of elliptic operators associated with discrete groups of quantized canonical transformations
• Noncommutative principal bundles through twist deformation
• A reconstruction theorem for Connes-Landi deformations of commutative spectral triples
• The ADHM Construction of Instantons on Noncommutative Spaces
• Vector bundles for "Matrix algebras converge to the sphere"
• Spin-Hall effect with quantum group symmetry
• Elliptic theory for operators associated with diffeomorphisms of smooth manifolds
• Part I, Free Actions of Compact Abelian Groups on C*-Algebras
• Noncommutative elliptic theory. Examples
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• On Noncommutative Principal Bundles with Finite Abelian Structure Group
• Lifting spectral triples to noncommutative principal bundles
• Factorization of Dirac operators on toric noncommutative manifolds
• Differential and Twistor Geometry of the Quantum Hopf Fibration
• Braided Hopf algebras and gauge transformations II: $\ast$-structures and examples
• Finitely-Generated Projective Modules over the Theta-deformed 4-sphere
• Deformation quantization of principal bundles
• Part III, Free Actions of Compact Quantum Groups on C*-Algebras
• On the Relationship between Classical and Deformed Hopf Fibrations
• An Atiyah Sequence for Noncommutative Principal Bundles
• The Noncommutative Topology of Anti-Self-Dual Gauge Fields