. Dolan-grady-est-introduite-etétudiéeetétudiée, Pour les autres cas q = 1, un homomorphisme d'algèbre explicite avec une sous algèbre coidéale de U q ( g) est présenté. Les théories de champs quantiques de Toda avec bords sont ensuite considérées. Pour lapremì ere fois, des relations algébriques pour la symétrie non-Abelienne sont obtenues explicitement Comme conséquence, ` a partir de considérations purement algébriques, les contraintes suffisantes pour toutes les conditions aux bords intégrables (fixe ou dynamique) sont classifiées. Il est nécessaire de donner quelques précisions sur les notations de ce papier. Le premier point correspondàcorrespondà l'appellation " q?Onsager " plutôt que " q?Dolan-Grady " de l'algèbre. Afin de clarifier la compréhension le nom " q?Dolan-Grady " a ´ eté utilisé dans cette thèse pour bien insister sur le lien avec le cas q = 1. Cependant, lors de l'introduction de O q ( sl 2 ) par Terwilliger, Cet article suit donc cette convention. Le second point est l'utilisation de minusculè a la place de majuscule pour désigner les algèbres affines. L'article est actuellement soumisàsoumisà " Letters in Mathematical Physics

L. Dépendant-deséquationsdeséquations-de-bethe, avons pas donc totalement résolu le probì eme spectral Il restè a résoudre leséquationsleséquations de Bethe Dans le cas des cha??nescha??nes de spins universelles, cette résolution n'existe pas. Il existe cependant de nombreuses solutions au cas par cas dans la littérature (voir références dans Lorsque le groupe quantique est le Yangien de gl(n), une solution pour les cha??nescha??nes dites L 0 périodiques (la cha??necha??ne est définie par séquence de L 0 spins arbitraires se répétant) a ´ eté formulée et permet d'obtenir l'´ etat fondamental ainsi que les petites excitations au-dessus du vide, Leprobì eme similaire pour l'algèbre quantique, les super-groupes quantiques et les cas avec bords reste ouvert, 0198.

. Enfin, nous avons considéré leprobì eme des modèles intégrables quantiques avec bords

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