Quantum symmetry groups of noncommutative tori

TL;DR

This paper constructs and analyzes a family of compact quantum groups acting on noncommutative tori, describing their structure, representations, and Haar measure, extending classical symmetry groups to the quantum setting.

Contribution

It introduces a new family of quantum symmetry groups for noncommutative tori, detailing their algebraic structure and representation theory, and generalizing classical symmetry groups.

Findings

For θ=0, the quantum group reduces to the classical semidirect product of torus and symmetric group.

The algebra A_θ^n extends classical symmetry groups to the quantum setting.

Irreducible representations of the quantum groups correspond to those of the classical semidirect product.

Abstract

We discuss necessary conditions for a compact quantum group to act on the algebra of noncommutative $n$-torus $\mathbb{T}_\theta^n$ in a filtration preserving way in the sense of Banica and Skalski. As a result, we construct a family of compact quantum groups $\mathbb{G}_\theta=(A_\theta^n,\Delta)$ such that for each $\theta$, $\mathbb{G}_\theta$ is the final object in the category of all compact quantum groups acting on $\mathbb{T}_\theta^n$ in a filtration preserving way. We describe in details the structure of the C*-algebra $A_\theta^n$ and provide a concrete example of its representation in bounded operators. Moreover, we compute the Haar measure of $\mathbb{G}_\theta$. For $\theta=0$, the quantum group $\mathbb{G}_0$ is nothing but the classical group $\mathbb{T}^n\rtimes S_n$, where $S_n$ is the symmetric group. For general $\theta$, $\mathbb{G}_\theta$ is still an extension of the classical group $\mathbb{T}^n$ by the classical group $S_n$. It turns out that for $n=2$, the algebra $A_\theta^2$ coincides with the algebra of the quantum double-torus described by Hajac and Masuda. Using a variation of the little subgroup method we show that irreducible representations of $\mathbb{G}_\theta$ are in one-to-one correspondence with irreducible representations of $\mathbb{T}^n\rtimes S_n$.