Homogeneous quantum symmetries of finite spaces over the circle group

Abstract: Suppose $D$ is a finite dimensional C*-algebra carrying a continuous action $\overline{\Pi}$ of the circle group $\mathbb{T}$. We study the quantum symmetry group of $D$, taking $\overline{\Pi}$ into account. We show that they are braided compact quantum groups $\mathbb{G}$ over $\mathbb{T}$. Here, the R-matrix, $\mathbb{Z}\times\mathbb{Z}\ni (m,n)\to \zeta^{-m\cdot n}\in\mathbb{T}$ for a fixed $\zeta\in \mathbb{T}$, governs the braided structure. In particular, if $\overline{\Pi}$ is trivial, $\zeta=1$ or $D$ is commutative, then $\mathbb{G}$ coincides with Wang's quantum group of automorphisms of $D$. Moreover, we show that the bosonisation of $\mathbb{G}$ corresponds to the quantum symmetry group of the crossed product C*-algebra $D\rtimes\mathbb{Z}$, where the $\mathbb{Z}$-action is generated by $\overline{\Pi}_{\zeta{^{-1}}}$.