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. 

Quantum Groups and Noncommutative Complex Geometry.

Noncommutative Riemannian geometry is an area that has seen intense activity over the past 25 years. Despite this, noncommutative complex geometry is only now beginning to receive serious attention.

A Dolbeault-Dirac Spectral Triple for Quantum Projective Space.

The notion of a Kahler structure for a differential calculus was recently introduced by the second author as a framework in which to study the noncommutative geometry of the quantum flag manifolds.

THE QUANTUM FLAG MANIFOLD SUq(3)/T AS AN EXAMPLE OF A NONCOMMUTATIVE SPHERE BUNDLE

The quantum flag manifold SUq(3)/T is interpreted as a noncommtative bundle over the quantum complex projective plane with the quantum or Podleś sphere as a fibre. A connection arising from the

Q A ] 1 F eb 2 02 0 POSITIVE LINE BUNDLES OVER THE IRREDUCIBLE QUANTUM FLAG MANIFOLDS FREDY

Noncommutative Kähler structures were recently introduced by the third author as a framework for studying noncommutative Kähler geometry on quantum homogeneous spaces. It was subsequently observed

References

SHOWING 1-10 OF 61 REFERENCES

Noncommutative Geometry and Quantum Group Symmetries

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

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

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)

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

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

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

The homogeneous coordinate ring of the quantum projective plane

Quantum spheres

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

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
...