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

The common index characterizes the whole descent equation.

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.