We directly derive the crossing (fusion and braiding) matrices for arbitrary isospins of the WZW SU (2) model in the Feigin-Fuchs integral representation and prove explicttly that they are quantum Racah-Wigner 6j matrices under proper normalization conditions. For integer k of the Kac-Moody algebra, the matrices may be truncated and the admissible states ane chased under analytic continuation. The aon-admissible terms will non appear in the other channels after braiding. The consistency of crossing matrices with he ocalitv condition is also proven. Similar results are obrained for the minimal models, which are in agreement with the results obtained by G. Felder. J, Frdlich and G. Keller.

By uilizing the commution properties of the infinicesimal opcrators, We conserued the ortho-notmal bases of the inrrcucible represenzarions o group, Un expliciely witiout using any knowiedge abous the symmeatic group. Morcover, by using lowering opetators, a simple derivation of the representation of the infinitesimal opesators is aivea

This paper shows that, starting from any conserved current generated by some given infinitesimal symmetry generator, one may use flnite dual transtormations to induce infinitely many inflnitesimal symmetry generators. Thus, besdes starting from ordinary isotopic and space-time translation, this paper also disovers the infinitesimal generators for B

We extend the Gervais-Neveu-Felder equation (GNFE) to the boundary case. Using the bound-ary GNFE, we give a definition of the boundary elliptic quantum group BET,n (sl2). The evaluation representation of the boundary elliptic quantum group BET,n (sl2) is presented. The Bethe ansatz for the boundary quantum group BET.n (sl2) is obtained. In a special case, the obtained Bethe ansatz is similar to the Bethe ansatz of the eight-vertex and the higher spin eight-vertex models with open boundary conditions.

The relationship between Elliptic Ruijsenaars-Schneider (RS) and Calogero-Moser (CM) models with Sklyanin algebra is presented. Lax pair representations of the Elliptic RS and CM are reviewed. For n=2 case, the eigenvalue and eigenfunction for Lam