数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (6): 2573-2588.doi: 10.1007/s10473-023-0615-4

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JONES TYPE C*-BASIC CONSTRUCTION IN NON-EQUILIBRIUM HOPF SPIN MODELS*

Xiaomin WEI1, Lining JIANG2,†   

  1. 1. School of Physical and Mathematical Sciences, Nanjing Tech University, Nanjing 211816, China;
    2. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
  • 收稿日期:2022-05-10 修回日期:2023-05-31 发布日期:2023-12-08
  • 通讯作者: †Lining JIANG, E-mail: jianglining@bit.edu.cn
  • 作者简介:Xiaomin WEI, E-mail: wxiaomin@amss.ac.cn
  • 基金资助:
    This work was supported by the NSFC (11871303).

JONES TYPE C*-BASIC CONSTRUCTION IN NON-EQUILIBRIUM HOPF SPIN MODELS*

Xiaomin WEI1, Lining JIANG2,†   

  1. 1. School of Physical and Mathematical Sciences, Nanjing Tech University, Nanjing 211816, China;
    2. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
  • Received:2022-05-10 Revised:2023-05-31 Published:2023-12-08
  • Contact: †Lining JIANG, E-mail: jianglining@bit.edu.cn
  • About author:Xiaomin WEI, E-mail: wxiaomin@amss.ac.cn
  • Supported by:
    This work was supported by the NSFC (11871303).

摘要: Let H be a finite dimensional Hopf ${C}^*$-algebra, and let K be a Hopf *-subalgebra of H. Considering that the field algebra $\mathscr{F}_{K}$ of a non-equilibrium Hopf spin model carries a $D(H,K)$-invariant subalgebra $\mathscr{A}_{K}$, this paper shows that the ${C}^*$-basic construction for the inclusion $\mathscr{A}_{K} \subseteq \mathscr{F}_{K}$ {can be expressed as} the crossed product ${C}^*$-algebra $\mathscr{F}_{K} \rtimes D(H,K)$. Here, $D(H,K)$ is a bicrossed product of the opposite dual $\widehat{H^{op}}$ and $K$. Furthermore, the natural action of $\widehat{D(H,K)}$ on $D(H,K)$ gives rise to the iterated crossed product $\mathscr{F}_{K} \rtimes D(H,K) \rtimes \widehat{D(H,K)}$, which coincides with the ${C}^*$-basic construction for the inclusion $\mathscr{F}_{K} \subseteq \mathscr{F}_{K} \rtimes D(H,K)$. In the end, the Jones type tower of field algebra $\mathscr{F}_{K}$ is obtained, and the new field algebra emerges exactly as the iterated crossed product.

关键词: field algebra, conditional expectation, basic construction, ${C}^*$-tower

Abstract: Let H be a finite dimensional Hopf ${C}^*$-algebra, and let K be a Hopf *-subalgebra of H. Considering that the field algebra $\mathscr{F}_{K}$ of a non-equilibrium Hopf spin model carries a $D(H,K)$-invariant subalgebra $\mathscr{A}_{K}$, this paper shows that the ${C}^*$-basic construction for the inclusion $\mathscr{A}_{K} \subseteq \mathscr{F}_{K}$ {can be expressed as} the crossed product ${C}^*$-algebra $\mathscr{F}_{K} \rtimes D(H,K)$. Here, $D(H,K)$ is a bicrossed product of the opposite dual $\widehat{H^{op}}$ and $K$. Furthermore, the natural action of $\widehat{D(H,K)}$ on $D(H,K)$ gives rise to the iterated crossed product $\mathscr{F}_{K} \rtimes D(H,K) \rtimes \widehat{D(H,K)}$, which coincides with the ${C}^*$-basic construction for the inclusion $\mathscr{F}_{K} \subseteq \mathscr{F}_{K} \rtimes D(H,K)$. In the end, the Jones type tower of field algebra $\mathscr{F}_{K}$ is obtained, and the new field algebra emerges exactly as the iterated crossed product.

Key words: field algebra, conditional expectation, basic construction, ${C}^*$-tower

中图分类号: 

  • 16T05