Quantum symmetry and braid group statistics inG-spin models

Towards a quantum Galois theory for quantum double algebras of finite groups

“…This kind of classical statistical system or the corresponding quantum field theories will be called G-spin models [17]. The main motivation for studying such models is that they can provide the simplest example of quantum symmetry.…”

“…Using the inductive limit, A l can be extended to a C*-algebra A, called the field algebra of G-spin models (for details, see [17]). …”

“…The γ can be extended continuously to an action of D(G) on A, denote it by γ too, so that A is a D(G)-module algebra with respect to the γ ( [17]). Namely it is a bilinear map with properties that:…”

The duality theory of a finite dimensional discrete quantum group

“…In general, the G-spin models with an Abelian group G, are known to have a symmetry group G × G, where G denotes the Pontryagin dual of G (the group of characters of G). If G is non-Abelian, the Pontryagin dual G loses its meaning and the models have a symmetry of a double algebra D(G) [14], which is an algebra defined as the crossed product of C(G) and CG with the adjoint action of the latter on the former ( [2], [5]). In detail, letting F be the field algebra of a G-spin model [14], there is a natural action of D(G) on F so that F becomes a D(G)-modular algebra.…”

C*-Homomorphisms and duality of C*-discrete quantum groups

“…If there is a natural action of the double algebra D(G) on G so that G becomes a D(G)-algebra then the observable algebra O in G is obtained. Given an irreducible representation π of G , there emerges a realization of D(G) so that D(G) and π(O) are the commutants of each other [9]. Alongside May 7, 2007. this example, the Schur-Weyl duality between the symmetric group and the general linear group [10], the Jimbo-Schur-Weyl duality between a quantum group of type A and a Hecke algebra [11,12], and so on, fit into the scheme.…”