ON THE QUANTUM SYMMETRY OF THE CHIRAL ISING-MODEL

We introduce the notion of rational Hopf algebras that we think are able to describe the superselection symmetries of rational quantum field theories. As an example we show that a six-dimensional rational Hopf algebra H can reproduce the fusion rules, the conformal weights, the quantum dimensions and the representation of the modular group of the chiral Ising model. H plays the role of the global symmetry algebra of the chiral Ising model in the following sense: (1) a simple field algebra F and a representation pi on H(pi) of it is given, which contains the c = 1/2 unitary representations of the Virasoro algebra as subrepresentations; (2) the embedding U: H --> B(H(pi)) is such that the observable algebra pi(A)- is the invariant subalgebra of B(H(pi)) with respect to the left adjoint action of H and U(H) is the commutant of pi(A); (3) there exist H-covariant primary fields in B(H(pi)), which obey generalized Cuntz algebra properties and intertwine between the inequivalent sectors of the observables.