数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (3): 907-924.doi: 10.1007/s10473-021-0317-8

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THE FIELD ALGEBRA IN HOPF SPIN MODELS DETERMINED BY A HOPF *-SUBALGEBRA AND ITS SYMMETRIC STRUCTURE

魏晓敏1, 蒋立宁1, 辛巧玲2   

  1. 1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China;
    2. School of Mathematical Sciences, Tianjin Normal University, Tianjin 300387, China
  • 收稿日期:2020-02-26 修回日期:2020-04-20 出版日期:2021-06-25 发布日期:2021-06-07
  • 通讯作者: Lining JIANG E-mail:jianglining@bit.edu.cn
  • 作者简介:Xiaomin WEI,E-mail:wxiaomin1509@163.com; Qiaoling XIN,E-mail:xinqiaoling0923@163.com
  • 基金资助:
    The project is supported by National Nature Science Foundation of China (11871303, 11701423) and Nature Science Foundation of Hebei Province (A2019404009).

THE FIELD ALGEBRA IN HOPF SPIN MODELS DETERMINED BY A HOPF *-SUBALGEBRA AND ITS SYMMETRIC STRUCTURE

Xiaomin WEI1, Lining JIANG1, Qiaoling XIN2   

  1. 1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China;
    2. School of Mathematical Sciences, Tianjin Normal University, Tianjin 300387, China
  • Received:2020-02-26 Revised:2020-04-20 Online:2021-06-25 Published:2021-06-07
  • Contact: Lining JIANG E-mail:jianglining@bit.edu.cn
  • About author:Xiaomin WEI,E-mail:wxiaomin1509@163.com; Qiaoling XIN,E-mail:xinqiaoling0923@163.com
  • Supported by:
    The project is supported by National Nature Science Foundation of China (11871303, 11701423) and Nature Science Foundation of Hebei Province (A2019404009).

摘要: Denote a finite dimensional Hopf $C^*$-algebra by $H$, and a Hopf $*$-subalgebra of $H$ by $H_{1}$. In this paper, we study the construction of the field algebra in Hopf spin models determined by $H_{1}$ together with its symmetry. More precisely, we consider the quantum double $D(H,H_{1})$ as the bicrossed product of the opposite dual $\widehat{H^{op}}$ of $H$ and $H_{1}$ with respect to the coadjoint representation, the latter acting on the former and vice versa, and under the non-trivial commutation relations between $H_{1}$ and $\widehat{H}$ we define the observable algebra $\mathcal{A}_{H_{1}}$. Then using a comodule action of $D(H,H_{1})$ on $\mathcal{A}_{H_{1}}$, we obtain the field algebra $\mathcal{F}_{H_{1}}$, which is the crossed product $\mathcal{A}_{H_{1}} \rtimes \widehat{D(H,H_{1})}$, and show that the observable algebra $\mathcal{A}_{H_{1}}$ is exactly a $D(H,H_{1})$-invariant subalgebra of $\mathcal{F}_{H_{1}}$. Furthermore, we prove that there exists a duality between $D(H,H_{1})$ and $\mathcal{A}_{H_{1}}$, implemented by a $*$-homomorphism of $D(H,H_{1})$.

关键词: Comodule algebra, field algebra, observable algebra, commutant, duality

Abstract: Denote a finite dimensional Hopf $C^*$-algebra by $H$, and a Hopf $*$-subalgebra of $H$ by $H_{1}$. In this paper, we study the construction of the field algebra in Hopf spin models determined by $H_{1}$ together with its symmetry. More precisely, we consider the quantum double $D(H,H_{1})$ as the bicrossed product of the opposite dual $\widehat{H^{op}}$ of $H$ and $H_{1}$ with respect to the coadjoint representation, the latter acting on the former and vice versa, and under the non-trivial commutation relations between $H_{1}$ and $\widehat{H}$ we define the observable algebra $\mathcal{A}_{H_{1}}$. Then using a comodule action of $D(H,H_{1})$ on $\mathcal{A}_{H_{1}}$, we obtain the field algebra $\mathcal{F}_{H_{1}}$, which is the crossed product $\mathcal{A}_{H_{1}} \rtimes \widehat{D(H,H_{1})}$, and show that the observable algebra $\mathcal{A}_{H_{1}}$ is exactly a $D(H,H_{1})$-invariant subalgebra of $\mathcal{F}_{H_{1}}$. Furthermore, we prove that there exists a duality between $D(H,H_{1})$ and $\mathcal{A}_{H_{1}}$, implemented by a $*$-homomorphism of $D(H,H_{1})$.

Key words: Comodule algebra, field algebra, observable algebra, commutant, duality

中图分类号: 

  • 16T05