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In this paper we study the algebra of quantum observables of the Chern-Simonsmatrix model which was originally proposed by Susskind and Polychronakos todescribe electrons in fractional quantum Hall effects. We establish thecommutation relations for its generators and study the large $N$ limit of itsrepresentation. We show that the large $N$ limit algebra is isomorphic to theuniform in $N$ algebra studied by Costello, which is isomorphic to the deformeddouble current algebra studied by Guay. Under appropriate scaling limit, weshow that the large $N$ limit algebra degenerates to a Lie algebra which admitsa surjective map to the affine Lie algebra of $\mathfrak{u}(p)$. This leads toa complete proof of the large $N$ emergence of the $\mathfrak{u}(p)$ currentalgebra as proposed by Dorey, Tong and Turner. This also suggests a rigorousderivation of edge excitation of a fractional quantum Hall droplet.

Quantum Algebra of Chern-Simons Matrix Model and Large $N$ Limit