Emerging integrable structures  in Abelian and non-Abelian quantum Hall liquids
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I'll  review the main properties of  Laughlin incompressible quantum liquids. I
will discuss in particular   the low-energy edge excitations and  their
connection   to the area-preserving transformations. The corresponding
algebraic structures  suggest a way to go beyond the linear approximation
(Luttinger) theory for edge states . In this respect  I'll explain why   the
integrable 1D quantum model  Calogero-Sutherland plays a crucial role.
Recently, by using conformal field theory techniques, we have shown that a
similar  scenario is emerging  in the case of the non-Abelian quantum Hall
liquids. I will discuss the main results trying to focus only  the general
ideas behind this approach.