Non-commutative algebra of functions of 4-dimensional quantum Hall droplet

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 <f>S4</f>, which is the base of the bundle <f>S7</f> obtained by the second Hopf fibration, i.e., <f>S7/S3=S4</f>, appears naturally in this theory. The fuzzy monopole harmonics, which are the essential elements in the non-commutative algebra of functions on <f>S4</f>, 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 <f>SO(5)</f> angular momentum. In particular, the explicit expression of the coupling coefficients, in the theory of <f>SO(5)</f> 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. [Copyright &y& Elsevier]