Jones Type Basic Construction On G-spin Models

The main aim of this thesis is to study some properties about the field algebra and the observable algebra determined by a normal subgroup in G-spin models,including the internal symmetry,the specific construction of the observable algebra,the C*-index,the C*-basic construction and so on.This thesis consists of the following seven chapters.In Chapter 1,we present a general survey of the backgrounds and modern developments for Jones index and quantum spin models.Subsequently we show the important conclusions of this thesis,and related results for Hopf algebras,which will be used throughout this thesis.In Chapter 2,one establishes the quantum spin models theory determined by a normal subgroup in G-spin models.Firstly,the concept of the quantum double D(H;G)determined by a normal subgroup H is defined.It is shown that D(H;G)is only a subalgebra of D(G),the quantum double of G,but not a Hopf subalgebra of D(G).Later,the notion of the field algebra FH related to this model is given,which is a C*-subalgebra of the field algebra F of G-spin models.Secondly,one can define an action of D(H;G)on the field algebra FH,such that FH becomes a D(H;G)-module algebra,whereas F is not.Then the observable algebra A(H,G)determined by a normal subgroup H is obtained as the D(H;G)-invariant subspace of FH.Finally,one can give the symmetric structure in quantum spin models determined by a normal subgroup H from the point of view of representation theory.In Chapter 3,we mainly study the specific construction of the observable algebra de-termined by a normal subgroup H in G-spin models.Firstly,the concrete structure of the observable algebra A(H,G)is given.Secondly,using the twisted tensor product,one can con-struct a C*-algebra B ?…×H×G×H×G×H×… where G denotes the algebra of complex functions on G,and H the group algebra.Finally,one can verify that the observable algebra A(H,G)is C*-isomorphic to 3.In Chapter 4,one proves that the conditional expectation zH from the field algebra FH onto the observable algebra A(H,G)is of finite type.First of all,the observable algebra A(H,G)can be expressed as the C*-inductive limit for a group of C*-algebras,and then one can find out a quasi-basis for the conditional expectation zH:FH? A(H,G)thus the C*-index of the conditional expectation zH is |G||H|I.In Chapter 5,we present the concrete representation of the basic construction and C*-basic construction for the quantum double D(G)and its Hopf subalgebra D(G;H).Firstly,for algebras D(G;H)(?)D C D(G)and a conditional expectation ?:D(G)? D(G;H),we call the algebra(D(G),e)generated by D(G)and e the basic construction,where we regard ? as an idempotent element e on D(G).Secondly,one can define an action of CG on C(G/H ×G),such that this action defines an automorphic action,and then can construct a crossed product algebra C(G/H x G)x CG.Finally,one can show that the basic construction(D(G),e)is algebra isomorphic to the crossed product algebra C(G/H x G)x CG.Similarly,we can construct a crossed product C*-algebra C(G/H xG)x CG,and then prove that the C*-basic construction C*<D(G),e)for the quantum double D(G)is C*-isomorphic to the crossed product C*-algebra C(G/H x G)x CG.In Chapter 6,we establish Jones type basic construction on the field algebra of G-spin models.Firstly,since F is a D(G)-module algebra,one can construct a crossed product C*-algebra F × D(G)such that it is C*-isomorphic to the C*-basic construction for the field algebra F and the D(G)-invariant subalgebra.Secondly,under the defined D(G)-module action on F × D(G),an iterated crossed product C*-algebra can be obtained,which coincides with the C*-basic construction for the crossed product C*-algebra F × D(G)and the field algebra F.Finally,one can show that the iterated crossed product C*-algebra is a new field algebra and give the concrete form with the order operators and disorder operators.In Chapter 7,we summarize the main contribution of this thesis and makes the expec-tation for the further study.