Isomorphism classes for quantum Heisenberg manifolds

TL;DR

This paper classifies the isomorphism classes of irrational quantum Heisenberg manifolds by analyzing their K-theoretic invariants and group actions, providing a complete characterization based on parameters and symmetries.

Contribution

It establishes a complete classification of irrational quantum Heisenberg manifolds up to isomorphism using K-theory and group actions, linking algebraic parameters to geometric orbits.

Findings

All tracial states induce the same homomorphism on K_0 in the irrational case

Two irrational quantum Heisenberg manifolds are isomorphic iff their parameters are in the same GL_2(Z) orbit

The embedding into a crossed product algebra facilitates the classification

Abstract

We embed the quantum Heisenberg manifold in a crossed product algebra. This enables us to show that, in the irrational case, all tracial states on $\dc$ induce the same homomorphism on the K_0-group. We conclude that two irrational quantum Heisenberg manifolds $\dc$ and $D^c_{\mu ' \nu '}$ are isomorphic if and only if the parameters $(\mu,\nu)$ and $(\mu ',\nu ')$ belong to the same orbit under the usual action of $GL_2(\ZZ)$ on the torus.