Noncommutative Coverings of Quantum Tori

TL;DR

The paper develops a framework for noncommutative coverings of quantum tori by free actions of finite groups on unital C$^*$-algebras, aiming to define a fundamental-group–like invariant for noncommutative spaces. It shows that connected coverings of a quantum torus $\mathbb{A}^n_{\theta}$ (with $\theta$ quite irrational) are classified by data $(M,\theta')$ yielding a gauge action by $G \cong (M^{-1}\mathbb{Z}^n)/\mathbb{Z}^n$ and a transformed matrix $\theta'$ satisfying $M\theta'M^T \in \theta + M_n(\mathbb{Z})$, leading to abelian, finite-index structure groups and a profinite fundamental group $\pi^n_{\theta}$. The paper further extends to smooth coverings of $\mathbb A^2_{\theta}$ by finite Abelian groups, constructing coverings from Picard homomorphisms into $\mathrm{Out}^\infty(\mathbb A^2_{\theta})$ via Bogoliubov-type unitaries on $L^2(\mathbb{R})$ and uniting isotypic components into a coherent $\hat{G}$-graded C$^*$-algebra. Collectively, these results align noncommutative coverings with classical torus coverings, connect to Morita theory through Picard groups, and lay groundwork toward a noncommutative fundamental group for quantum tori.

Abstract

We introduce a framework for coverings of noncommutative spaces. Moreover, we study noncommutative coverings of irrational quantum tori and characterize all such coverings that are connected in a reasonable sense.

Theorems & Definitions (25)

• Definition 4.1
• Remark 4.2
• Lemma 4.3
• proof
• Lemma 4.4
• proof
• Remark 4.5
• Theorem 4.6
• proof
• Remark 4.7