QUANTUM SPIN CHAINS WITH QUANTUM GROUP SYMMETRY

We consider actions of quantum groups on lattice spin systems. We show that if an action of a quantum group respects the local structure of a lattice system, it has to be an ordinary group. Even allowing weakly delocalized (quasi-local) tails of the action, we find that there are no actions of a properly quantum group commuting with lattice transla tions. The non-locality arises from the ordering of factors in the qua ntum group C-algebra, and can be made one-sided, thus allowing semi-l ocal actions on a half chain. Under such actions, localized quantum gr oup invariant elements remain localized. Hence the notion of interacti ons invariant under the quantum group and also under translations, rec ently studied by many authors, makes sense even though there is no glo bal action of the quantum group. We consider a class of such quantum g roup invariant interactions with the property that there is a unique t ranslation invariant ground state. Under weak locality assumptions, it s GNS representation carries no unitary representation of the quantum group.