Discrete quantum group coaction on the circle

TL;DR

The paper asks whether discrete quantum groups can provide genuine symmetry of the circle under a weakly faithful, linear coaction on $C(S^1)$. Using the universal discrete quantum automorphism framework and Tannaka-Krein duality, it reduces possible actions to a finite-dimensional, matrix-encoded setting and shows that the resulting algebra must be commutative. The main result is that any Kac-type DQG coacting weakly faithfully and linearly on $C(S^1)$ is classical, i.e., isomorphic to $C_0(\Gamma)$ for some discrete group $\Gamma$. This extends non-existence results for genuine quantum symmetries from compact spaces to discrete quantum groups acting on simple manifolds and provides a method potentially extensible to broader classes of manifolds.

Abstract

We prove by an explicit calculation that if any Kac-type $C^*$-algebraic discrete quantum group $\mathcal{Q}$ has a `weakly faithful' coaction on $C(S^1)$ which is `linear' in the sense that it leaves the space spanned by $\{ Z, \overline{Z} \}$ invariant, then $\mathcal{Q}$ must be classical, i.e. isomorphic with $C_0(Γ)$ for some discrete group $Γ$. This parallels the well-known result of non-existence of genuine compact quantum group symmetry obtained by the first author and his collaborators ([32] and the references therein).

Theorems & Definitions (12)

• Definition 3.1
• Remark 3.2
• Theorem 3.3
• Definition 3.4
• Theorem 3.5
• Theorem 3.6
• Theorem 3.7
• Definition 4.1
• Theorem 4.2
• Theorem 4.3