Let H be a finite dimensional Hopf C^(*)-algebra,and let K be a Hopf^(*)-subalgebra of H.Considering that the field algebra■K of a non-equilibrium Hopf spin model carries a D(H,K)-invariant subalgebra ■K,this paper ...Let H be a finite dimensional Hopf C^(*)-algebra,and let K be a Hopf^(*)-subalgebra of H.Considering that the field algebra■K of a non-equilibrium Hopf spin model carries a D(H,K)-invariant subalgebra ■K,this paper shows that the C^(*)-basic construction for the inclusion ■K×■K can be expressed as the crossed product C^(*)-algebra■KD(H,K).Here,D(H,K)is a bicrossed product of the opposite dual H^(op) and K.Furthermore,the natural action of D(H,K)on D(H,K)gives rise to the iterated crossed product■KD(H,K)×D(H,K),which coincides with the C^(*)-basic construction for the inclusion■K×■KD(H,K).In the end,the Jones type tower of field algebra■Kis obtained,and the new field algebra emerges exactly as the iterated crossed product.展开更多
Let H be a finite Hopf C^＊ -algebra and H′be its dual Hopf algebra. Drinfeld＇s quantum double D（H） of H is a Hopf^＊-algebra. There is a faithful positive linear functional θ on D（H）. Through the associated Ge...Let H be a finite Hopf C^＊ -algebra and H′be its dual Hopf algebra. Drinfeld＇s quantum double D（H） of H is a Hopf^＊-algebra. There is a faithful positive linear functional θ on D（H）. Through the associated Gelfand-Naimark-Segal （GNS） representation, D（H） has a faithful^＊ -representation so that it becomes a Hopf C^＊ -algebra. The canonical embedding map of H into D（H） is isometric.展开更多
Let F be the field algebra of G -spin model, D(G) the double algebra of a finite group G and D(H) the sub-Hopf algerba of D(G) determined by the subgroup H of G . The paper builds a correspondence between D(H) and th...Let F be the field algebra of G -spin model, D(G) the double algebra of a finite group G and D(H) the sub-Hopf algerba of D(G) determined by the subgroup H of G . The paper builds a correspondence between D(H) and the D(H) -invariant sub- C * -algebra A H in F, and proves that the correspondence is strictly monotonic.展开更多
Let A be a commutative C^＊ -algebra. By the Gelfand-Naimark theorem, there exists a locally compact space G such that A is isomorphic to Co（G）, the C^＊-algebra of all complex continuous functions on G vanishing at...Let A be a commutative C^＊ -algebra. By the Gelfand-Naimark theorem, there exists a locally compact space G such that A is isomorphic to Co（G）, the C^＊-algebra of all complex continuous functions on G vanishing at infinity. The result is generalized to the ease of Hopf C^＊-algebra, where G is altered by a locally compact group. Using the isomorphic representation, the counit ε and the antipode S of a commutative Hopf C^＊-algebra are proposed.展开更多
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...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, H_(1)) as the bicrossed product of the opposite dual Hopˆ 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,H_(1)), 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, H_(1)).展开更多
文摘Let H be a finite dimensional Hopf C^(*)-algebra,and let K be a Hopf^(*)-subalgebra of H.Considering that the field algebra■K of a non-equilibrium Hopf spin model carries a D(H,K)-invariant subalgebra ■K,this paper shows that the C^(*)-basic construction for the inclusion ■K×■K can be expressed as the crossed product C^(*)-algebra■KD(H,K).Here,D(H,K)is a bicrossed product of the opposite dual H^(op) and K.Furthermore,the natural action of D(H,K)on D(H,K)gives rise to the iterated crossed product■KD(H,K)×D(H,K),which coincides with the C^(*)-basic construction for the inclusion■K×■KD(H,K).In the end,the Jones type tower of field algebra■Kis obtained,and the new field algebra emerges exactly as the iterated crossed product.
文摘Let H be a finite Hopf C^＊ -algebra and H′be its dual Hopf algebra. Drinfeld＇s quantum double D（H） of H is a Hopf^＊-algebra. There is a faithful positive linear functional θ on D（H）. Through the associated Gelfand-Naimark-Segal （GNS） representation, D（H） has a faithful^＊ -representation so that it becomes a Hopf C^＊ -algebra. The canonical embedding map of H into D（H） is isometric.
文摘Let F be the field algebra of G -spin model, D(G) the double algebra of a finite group G and D(H) the sub-Hopf algerba of D(G) determined by the subgroup H of G . The paper builds a correspondence between D(H) and the D(H) -invariant sub- C * -algebra A H in F, and proves that the correspondence is strictly monotonic.
文摘Let A be a commutative C^＊ -algebra. By the Gelfand-Naimark theorem, there exists a locally compact space G such that A is isomorphic to Co（G）, the C^＊-algebra of all complex continuous functions on G vanishing at infinity. The result is generalized to the ease of Hopf C^＊-algebra, where G is altered by a locally compact group. Using the isomorphic representation, the counit ε and the antipode S of a commutative Hopf C^＊-algebra are proposed.
基金supported by National Nature Science Foundation of China(11871303,11701423)Nature Science Foundation of Hebei Province(A2019404009)。
文摘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, H_(1)) as the bicrossed product of the opposite dual Hopˆ 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,H_(1)), 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, H_(1)).
