YANGIANS, INTEGRABLE QUANTUM-SYSTEMS AND DOREYS RULE

It was pointed out by P. Dorey that the three-point couplings between the quantum particles in affine Toda field theories have a remarkable Lie-theoretic interpretation. It is also well known that such theories admit quantum affine algebras as ''quantum symmetry groups,'' and wid ely believed that the quantum particles correspond to the so-called fu ndamental representations of these algebras. This led to the conjectur e that Dorey's rule should describe when a fundamental representation occurs with non-zero multiplicity in a tensor product of two other fun damental representations. The purpose of this paper is to prove this c onjecture, both for quantum affine algebras and for Yangians. The resu lt reveals a hitherto unsuspected role played by Coxeter elements (and their twisted analogues) in the representation theory of these algebr as.