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

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

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

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

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

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doi:10.1007/s10473-023-0615-4

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

Xiaomin WEI¹, Lining JIANG^2,†

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).

Xiaomin WEI¹, Lining JIANG^2,†

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

Xiaomin WEI, Lining JIANG. JONES TYPE C^*-BASIC CONSTRUCTION IN NON-EQUILIBRIUM HOPF SPIN MODELS^*[J]. 数学物理学报(英文版), 2023, 43(6): 2573-2588.

Xiaomin WEI, Lining JIANG. JONES TYPE C^*-BASIC CONSTRUCTION IN NON-EQUILIBRIUM HOPF SPIN MODELS^*[J]. Acta mathematica scientia,Series B, 2023, 43(6): 2573-2588.

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