Quantum teardrops

TL;DR

This paper introduces and analyzes the algebraic and topological structures of quantum weighted projective lines, including their representations, bundle structures, and K-theory, expanding the understanding of noncommutative geometric spaces.

Contribution

It provides a detailed presentation of the coordinate algebra of quantum teardrops, classifies their irreducible *-representations, and explores their bundle and K-theoretic properties.

Findings

Explicit generators and relations for quantum teardrops

Classification of irreducible *-representations

Computation of K-groups for associated C*-algebras

Abstract

Algebras of functions on quantum weighted projective spaces are introduced, and the structure of quantum weighted projective lines or quantum teardrops are described in detail. In particular the presentation of the coordinate algebra of the quantum teardrop in terms of generators and relations and classification of irreducible *-representations are derived. The algebras are then analysed from the point of view of Hopf-Galois theory or the theory of quantum principal bundles. Fredholm modules and associated traces are constructed. C*-algebras of continuous functions on quantum weighted projective lines are described and their K-groups computed.