  We discuss the bosonization of nonrelativistic fermions interacting with
non-Abelian gauge fields in the lowest Landau level in the framework of higher
dimensional quantum Hall effect. The bosonic action is a one-dimensional matrix
action, which can also be written as a noncommutative field theory, invariant
under $W_N$ transformations. The requirement that the usual gauge
transformation should be realized as a $W_N$ transformation provides an analog
of a Seiberg-Witten map, which allows us to express the action purely in terms
of bosonic fields. The semiclassical limit of this, describing the gauge
interactions of a higher dimensional, non-Abelian quantum Hall droplet,
produces a bulk Chern-Simons type term whose anomaly is exactly cancelled by a
boundary term given in terms of a gauged Wess-Zumino-Witten action.
