# Geometry of Quantum Projective Spaces

@article{DAndrea2012GeometryOQ, title={Geometry of Quantum Projective Spaces}, author={Francesco D’Andrea and Giovanni Landi}, journal={arXiv: Quantum Algebra}, year={2012} }

In recent years, several quantizations of real manifolds have been studied, in particular from the point of view of Connes' noncommutative geometry. Less is known for complex noncommutative spaces. In this paper, we review some recent results about the geometry of complex quantum projective spaces.

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## References

SHOWING 1-10 OF 61 REFERENCES

### Noncommutative Geometry and Quantum Group Symmetries

- Mathematics
- 2008

Quantum groups and quantum homogeneous spaces - developed by several authors since the 80's - provide a large class of examples of algebras which for many reasons we interpret as `coordinate…

### Geometry of the quantum projective plane

- Physics, Mathematics
- 2009

We review some of the geometry of the quantum projective plane with emphasis on the construction of a differential calculus and of the Dirac operator (of a spin^c-structure). We also report on…

### Bounded and unbounded Fredholm modules for quantum projective spaces

- Mathematics
- 2010

We construct explicit generators of the K-theory and K-homology of the coordinate algebras of functions on the quantum projective spaces. We also sketch a construction of unbounded Fredholm modules,…

### Geometry of the quantum complex projective spaceCPq(N)

- Physics, Mathematics
- 1996

The quantum deformationCPq(N) of complex projective space is discussed. Many of the features present in the case of the quantum sphere can be extended. The differential and integral calculus is…

### An Introduction to Noncommutative Spaces and Their Geometries

- Mathematics
- 1997

Noncommutative Spaces and Algebras of Functions.- Projective Systems of Noncommutative Lattices.- Modules as Bundles.- A Few Elements of K-Theory.- The Spectral Calculus.- Noncommutative Differential…

### Bundles over Quantum Sphere and Noncommutative Index Theorem

- Mathematics
- 2000

The Noncommutative Index Theorem is used to prove that the Chern numbers of quantum Hopf line bundles over the standard Podles quantum sphere equal the winding numbers of the repres- entations…

### Quantum spheres

- Physics
- 1987

Spaces homogeneous under the action of the quantum SU(2) group are introduced and investigated. These spaces can be considered as noncommutative two-dimensional spheres of different radii. The…

### Conformal Structures in Noncommutative Geometry

- Mathematics
- 2007

It is well-known that a compact Riemannian spin manifold can be reconstructed from its canonical spectral triple which consists of the algebra of smooth functions, the Hilbert space of square…