Geometry of Quantum Projective Spaces

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.Comment: 48 pages, no figure

Non-Associative Geometry of Quantum Tori

We describe how to obtain the imprimitivity bimodules of the noncommutative torus from a "principal bundle" construction, where the total space is a quasi-associative deformation of a 3-dimensional Heisenberg manifold

Pimsner algebras and circle bundles

We report on the connections between noncommutative principal circle bundles, Pimsner algebras and strongly graded algebras. We illustrate several results with the examples of quantum weighted projective and lens spaces and theta-deformations.Comment: 24 pages. v3: Updated title. No changes in the scientific content and result

Quantum weighted projective and lens spaces

We generalize to quantum weighted projective spaces in any dimension previous results of us on K-theory and K-homology of quantum projective spaces `tout court'. For a class of such spaces, we explicitly construct families of Fredholm modules, both bounded and unbounded (that is spectral triples), and prove that they are linearly independent in the K-homology of the corresponding C*-algebra. We also show that the quantum weighted projective spaces are base spaces of quantum principal circle bundles whose total spaces are quantum lens spaces. We construct finitely generated projective modules associated with the principal bundles and pair them with the Fredholm modules, thus proving their non-triviality.Comment: 30 pages, no figures. Section on spectral triples expanded with some new result

Bounded and unbounded Fredholm modules for quantum projective spaces

We construct explicit generators of the K-theory and K-homology of the coordinate algebra of `functions' on quantum projective spaces. We also sketch a construction of unbounded Fredholm modules, that is to say Dirac-like operators and spectral triples of any positive real dimension.Comment: 8 pages, no figures, dcpic, pdflatex. v2: 8 pages, minor changes, final version to appears in JK

Deformations of the Canonical Commutation Relations and Metric Structures

Using Connes distance formula in noncommutative geometry, it is possible to retrieve the Euclidean distance from the canonical commutation relations of quantum mechanics. In this note, we study modifications of the distance induced by a deformation of the position-momentum commutation relations. We first consider the deformation coming from a cut-off in momentum space, then the one obtained by replacing the usual derivative on the real line with the h- and q-derivatives, respectively. In these various examples, some points turn out to be at infinite distance. We then show (on both the real line and the circle) how to approximate points by extended distributions that remain at finite distance. On the circle, this provides an explicit example of computation of the Wasserstein distance

Twist star products and Morita equivalence

We present a simple no-go theorem for the existence of a deformation quantization of a homogeneous space M induced by a Drinfel'd twist: we argue that equivariant line bundles on M with non-trivial Chern class and symplectic twist star products cannot both exist on the same manifold M. This implies, for example, that there is no symplectic star product on the complex projective spaces induced by a twist based on U(gl(n,C))[[h]] or any sub-bialgebra, for every n greater or equal than 2.Comment: 10 pages, no figure

Anti-selfdual Connections On The Quantum Projective Plane: Instantons

We study one-instantons, that is anti-selfdual connections with instanton number 1, on the quantum projective plane with orientation which is reversed with respect to the usual one. The orientation is fixed by a suitable choice of a basis element for the rank 1 free bimodule of top forms. The noncommutative family of solutions is foliated, each non-singular leaves being isomorphic to the quantum projective plane itself.Comment: 35 pages, no figures; v2: minor change