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.

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

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.

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

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.