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

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

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

Some remarks on K-lattices and the Adelic Heisenberg Group for CM curves

We define an adelic version of a CM elliptic curve $E$ which is equipped with an action of the profinite completion of the endomorphism ring of $E$. The adelic elliptic curve so obtained is provided with a natural embedding into the adelic Heisenberg group. We embed into the adelic Heisenberg group the set of commensurability classes of arithmetic $1$-dimensional $\mathbb{K}$-lattices (here and subsequently, $\mathbb{K}$ denotes a quadratic imaginary number field) and define theta functions on it. We also embed the groupoid of commensurability modulo dilations into the union of adelic Heisenberg groups relative to a set of representatives of elliptic curves with $R$-multiplication ($R$ is the ring of algebraic integers of $\mathbb{K}$). We thus get adelic theta functions on the set of $1$-dimensional $\mathbb{K}$-lattices and on the groupoid of commensurability modulo dilations. Adelic theta functions turn out to be acted by the adelic Heisenberg group and behave nicely under complex automorphisms (Theorems 6.12 and 6.14).Comment: 25 pages, no figures. Extensively revised version according to the comments of the reviewer

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

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

On noncommutative equivariant bundles

We discuss a possible noncommutative generalization of the notion of an equivariant vector bundle. Let $A$ be a $\mathbb{K}$-algebra, $M$ a left $A$-module, $H$ a Hopf $\mathbb{K}$-algebra, $\delta:A\to H\otimes A:=H\otimes_{\mathbb{K}} A$ an algebra coaction, and let $(H\otimes A)_\delta$ denote $H\otimes A$ with the right $A$-module structure induced by~$\delta$. The usual definitions of an equivariant vector bundle naturally lead, in the context of $\mathbb{K}$-algebras, to an $(H\otimes A)$-module homomorphism $$\Theta:H\otimes M\to (H\otimes A)_\delta\otimes_AM$$ that fulfills some appropriate conditions. On the other hand, sometimes an $(A,H)$-Hopf module is considered instead, for the same purpose. When $\Theta$ is invertible, as is always the case when $H$ is commutative, the two descriptions are equivalent. We point out that the two notions differ in general, by giving an example of a noncommutative Hopf algebra $H$ for which there exists such a $\Theta$ that is not invertible and a left-right $(A,H)$-Hopf module whose corresponding homomorphism $M\otimes H\to (A\otimes H)_\delta\otimes_AM$ is not an isomorphism.Comment: In this version we dismiss the term neb-homomorphism (hinting at 'noncommutative equivariant bundles'), as the class of modules is larger than the class of algebraic counterparts of vector bundles. We also corrected some mistakes. Our main example does not immediately extended to the left-right case and the example about the 'exotic' Hopf module works only in the left-right cas

Matrix Geometries Emergent from a Point

We describe a categorical approach to finite noncommutative geometries. Objects in the category are spectral triples, rather than unitary equivalence classes as in other approaches. This enables to treat fluctuations of the metric and unitary equivalences on the same footing, as representatives of particular morphisms in this category. We then show how a matrix geometry (Moyal plane) emerges as a fluctuation from one point, and discuss some geometric aspects of this space.Comment: 1 figur