Observable algebra in field algebra of G-spin models

Field algebra of G-spin models can provide the simplest examples of lattice field theory exhibiting quantum symmetry. Let D(G) be the double algebra of a finite group G and D(H), a sub-algebra of D(G) determined by subgroup H of G. This paper gives concrete generators and the structure of the observable algebra A H, which is a D(H)-invariant sub-algebra in the field algebra of G-spin models F, and shows that A
H
is a C
^∗
-algebra. The correspondence between H and A
H
is strictly monotonic. Finally, a duality between D(H) and A
H
is given via an irreducible vacuum C
^∗
-representation of F.