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 $D⋊\mathbb{Z}$, where the $\mathbb{Z}$-action is generated by ${\overline{\Pi }}_{\zeta {}^{-1}}$.

18 pages, added results on direct sum of matrix algebras and compared the framework with quantum symmetry groups coming from orthogonal filtrations