TY - JOUR

T1 - The Field Algebra in Hopf Spin Models Determined by a Hopf *-Subalgebra and Its Symmetric Structure

AU - Wei, Xiaomin

AU - Jiang, Lining

AU - Xin, Qiaoling

N1 - Publisher Copyright:
© 2021, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences.

PY - 2021/5

Y1 - 2021/5

N2 - Denote a finite dimensional Hopf C*-algebra by H, and a Hopf *-subalgebra of H by H1. In this paper, we study the construction of the field algebra in Hopf spin models determined by H1 together with its symmetry. More precisely, we consider the quantum double D(H, H1) as the bicrossed product of the opposite dual Hop^ of H and H1 with respect to the coadjoint representation, the latter acting on the former and vice versa, and under the non-trivial commutation relations between H1 and Ĥ we define the observable algebra AH1. Then using a comodule action of D(H, H1) on AH1, we obtain the field algebra ℱH1, which is the crossed product AH1⋊D(H,H1)^, and show that the observable algebra AH1 is exactly a D(H, H1)-invariant subalgebra of ℱH1. Furthermore, we prove that there exists a duality between D(H, H1) and AH1, implemented by a *-homomorphism of D(H, H1).

AB - Denote a finite dimensional Hopf C*-algebra by H, and a Hopf *-subalgebra of H by H1. In this paper, we study the construction of the field algebra in Hopf spin models determined by H1 together with its symmetry. More precisely, we consider the quantum double D(H, H1) as the bicrossed product of the opposite dual Hop^ of H and H1 with respect to the coadjoint representation, the latter acting on the former and vice versa, and under the non-trivial commutation relations between H1 and Ĥ we define the observable algebra AH1. Then using a comodule action of D(H, H1) on AH1, we obtain the field algebra ℱH1, which is the crossed product AH1⋊D(H,H1)^, and show that the observable algebra AH1 is exactly a D(H, H1)-invariant subalgebra of ℱH1. Furthermore, we prove that there exists a duality between D(H, H1) and AH1, implemented by a *-homomorphism of D(H, H1).

KW - 16T05

KW - 46L05

KW - 46N50

KW - 81R15

KW - Comodule algebra

KW - commutant

KW - duality

KW - field algebra

KW - observable algebra

UR - http://www.scopus.com/inward/record.url?scp=85104537410&partnerID=8YFLogxK

U2 - 10.1007/s10473-021-0317-8

DO - 10.1007/s10473-021-0317-8

M3 - Article

AN - SCOPUS:85104537410

SN - 0252-9602

VL - 41

SP - 907

EP - 924

JO - Acta Mathematica Scientia

JF - Acta Mathematica Scientia

IS - 3

ER -