Homogeneous quantum symmetries of finite spaces over the circle group

TL;DR

This paper investigates the quantum symmetry groups of finite-dimensional C*-algebras with circle group actions, revealing they are braided quantum groups influenced by an R-matrix, and connects these to automorphism groups and crossed product algebras.

Contribution

It introduces the concept of braided quantum symmetry groups over the circle group for finite-dimensional C*-algebras, extending Wang's automorphism quantum groups to a braided setting.

Findings

Quantum symmetry groups are braided compact quantum groups governed by an R-matrix.

When the action is trivial or the algebra is commutative, the symmetry group reduces to Wang's quantum automorphism group.

The bosonisation relates the quantum symmetry of the crossed product algebra to the original braided quantum 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 $\overline{\Pi}_{\zeta{^{-1}}}$.