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Classification of Abelian quantum Hall states and matrix formulation of topological fluids.

We give a simple and unified treatment of quantum topological fluids such as the quantum Hall fluid. We show that the order in such fluids can be characterized by a symmetric matrix K, in terms of which various physical quantities can be determined. We construct K by a matrix iteration procedure which may be decomposed into two simple elementary steps. The hierarchy construction is shown to be contained in our matrix iteration construction. The relationship between the vortex basis and the dual electron basis is clarified. We also show that under certain mild assumptions the generalized hierarchy construction exhausts all possible Abelian fractional quantum Hall states. We identify and determine the topological quantity known as the shift. Our formalism may be relevant for recent experimental data on multilayered systems.