Drinfel’d Twists and Algebraic Bethe Ansatz

We study representation theory of Drinfel’d twists, in terms of what we call F matrices, associated to finite dimensional irreducible modules of quantum affine algebras, and which factorize the corresponding (unitary) R-matrices. We construct explicitly such factorizing F -matrices for irreducible finite n tensor products of the fundamental evaluation representations of the quantum affine algebra Uq(ŝl2) and the Yangian Y (sl2). We then apply these constructions to the XXX1 2 and XXZ1 2 Heisenberg (inhomogeneous) quantum spin chains of finite lenght n in the framework of the Algebraic Bethe Ansatz. In particular, we show that these factorizing F -matrices diagonalize the generating matrix of scalar products of quantum states of these models. They also diagonalize the diagonal (operator) entries of the quantum monodromy matrix. Due to their algebraic properties, these F -matrices are shown to obey simple difference equations. This leads to a natural F -basis for the quantum space of states of the inhomogeneous XXX2 quantum spin chain of finite lenght in which this model can be interpreted as a diagonal dressing of the corresponding Gaudin model. URA 1436 ENSLAPP du CNRS, associée à l’Ecole Normale Supérieure de Lyon, à l’Université de Savoie et au Laboratoire d’Annecy de Physique des Particules. This work is supported by CNRS (France), the EC contract ERBCHRXCT920069 and grant PB93-0344 from DGICYT (Spain). email: [email protected]