Non-commutative Space, Two-dimensional Quantum Hall Fluid

At first we discuss how noncommutative geometry arises in quantum mechanics and formulates simple noncommutative field theory. Noncommutative field theory means that fields are thoughts of as function over noncommutative space. Then we formulates simple basic definition of fuzzy sphere, two kinds of quantum Hall effect and noncommutative Chern-Simons theory. In the same time, after shortly review the Halclane's description of the quantum Hall effect on 2-sphere S2, we obtain the corresponding noncommutative algebra An and the Moyal structure of the Hilbert space. By taking use of the method proposed by Susskind and Polychronakos, we give out a noncommutative Chern-Simons theory to describe the incompressible quantum Hall fluid on S2 and the related algebraic structure.