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

The common index characterizes the whole descent equation.

In accordance with the isospin direetton of some charge operator, we decompose the non-Abelian gauge potential and field strength explicitly and gauge covariantly into "neutral"fields, whose sources are these charges and their dual charges, and "vector particle" fieids which are charged. The deeomposition expressions of SU(N)/U(1)N-I, SU(N)/SU(P) SU(N-P) U(1), SO(N)/SO(N-1) are gives explicitly, including various generalized't Hooft field strength expressions. By separating variables of the equation of motion of the abelianizable gauge field,solution with static dual charge system and retarded solution of linearly accelerated dualcharge have been given.

Belavin's Zn

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.

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.

We derive the exchange relations of the vertex operators of Uq (A2(2)) and show that these vertex operators give the bosonization of the Izergin-Korepin model. We give an integral expression of the correlation functions of the Izergin-Korepin model and derive the difference equations which they satisfy.

We develop the description of non-commutative geometry of the 4-dimensional quantum Hall fluid's theory proposed recently by Zhang and Hu. The non-commutative structure of fuzzy S4, which is the base of the bundle S7 obtained by the second Hopf fibration, i.e., S7/S3=S4, appears naturally in this theory. The fuzzy monopole harmonics, which are the essential elements in the non-commutative algebra of functions on S4, are explicitly constructed and their obeying the matrix algebra is obtained. This matrix algebra is associative. We also propose a fusion scheme of the fuzzy monopole harmonics of the coupling system from those of the subsystems, and determine the fusion rule in such fusion scheme. By products, we provide some essential ingredients of the theory of SO(5) angular momentum. In particular, the explicit expression of the coupling coefficients, in the theory of SO(5) angular momentum, are given. We also discuss some possible applications of our results to the 4-dimensional quantum Hall system and the matrix brane construction in M-theory.

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