Outer action of representation category of discrete quantum groups: a case study

Abstract

Given any tracial von Neumann algebra $\mathcal{A}$, the first author has defined a unitary tensor functor $\Phi :\mathrm{Rep}({\mathcal{Q}}_{\mathrm{aut}}(\mathcal{A}))\to \mathrm{Bimod}(\mathcal{A})$ in his paper [Gos24]. In this paper, we consider $\mathcal{A}$ to be the underlying von Neumann algebra of $C^{\ast }(S_3)$ and show that given any DQG $\mathcal{Q}$, which contains $C^{\ast }(S_3)$ as a quantum subgroup and has an outer action $\tilde{\Phi }$ (in the sense of Definition 3.4}) on $\mathrm{Bimod}(\mathcal{A})$, which agrees with $\Phi$ when restricted on $\mathrm{Rep}(C^{\ast }(S_3))$, is monoidally equivalent to the DQG $C^{\ast }(S_3)$. Furthermore, given any DQG $\mathcal{Q}$ with an outer action $\Gamma :\mathrm{Rep}(\mathcal{Q})\to \mathrm{Bimod}(\mathcal{A})$, such that $\mathrm{Out}(\mathcal{A})\subseteq \mathrm{Im}(\Gamma )$, is either monoidally equivalent to Out$(\mathcal{A})$ or monoidally equivalent to $C^{\ast }(S_3)$ itself.

Preliminary version (14 pages + References). Comments are welcome