The Galois Correspondence in Field Algebra of G–spin Model

C⁎-Basic Construction from the Conditional Expectation on the Drinfeld Double

“…. One can prove ( ; ) becomes a Hopf * -algebra ( [15]). There exists a unique element = (1/| |) ∑ ℎ∈ ( , ℎ), called a cointegral, satisfying ∀ ∈ ( ; ), = = ( ) , and ( ) = 1.…”

The Quantum Symmetry in Nonbalanced Hopf Spin Models Determined by a Normal Coideal Subalgebra

“…e quantum symmetry in G-spin models generalizes the Z 2 × Z 2 symmetry which can sharply divide the ordered phase and the disordered phase in Ising models. Generally, if G is an Abelian group, G-spin models have a symmetry group G × G, which is the direct product of the group G and the group of characters of G. Based on a field-theory analysis of G-spin models, Jiang and Guo [2] gave the concrete construction of a D(G; N)-invariant subspace in field algebra of G-spin models and proved the D(G; N)-invariant subspace is Galois closed if N is a normal subgroup of G. On the contrary, Xin and Jiang [3] generalized G-spin models to G-spin models determined by a normal subgroup N, in which the quantum double D(N; G), and the field algebra determined by N are defined, and then, the observable algebra determined by N can be obtained as D(N; G)-invariant subalgebra. Based on these work, the quantum symmetry is given by the quantum double D(N; G).…”

“…(2) Suppose that G is a finite group and N is a normal subgroup of G. In this case of H � CG, the group algebra of G, K � CN can be viewed as a normal coideal * -subalgebra of H.Note that the quantum double D(G) of a finite group is the crossed product of C(G)…”

“…In order to check that U(X (1) )Π(F)U (SX (2) ) � Π(X(F)), it is enough to verify it on the generators Π(F H 1 )Ω of the Hilbert space H. For any…”