Quantum symmetry groups of noncommutative tori

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 ${\mathbb{T}}^n⋊S_n$, 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 ${\mathbb{T}}^n⋊S_n$.

25 pages, subsection 4.3 reedited, Propositions 4.2 and 4.4 removed, Remark 5.1 reedited

References in corpus (4)

• Harmonic analysis on quantum tori
• Quantum Group of Isometries in Classical and Noncommutative Geometry
• Quantum symmetry groups of C*-algebras equipped with orthogonal filtrations
• Equivalent Notions of Normal Quantum Subgroups, Compact Quantum Groups with Properties F and FD, and Other Applications