  In the previous work, it was shown that the degrees of freedom on the horizon
of BTZ black hole can be described by two chiral massless scalar fields with
opposite chirality. In this paper, we continuous this research. It is found
that the scalar field is actually a compact boson field on a circle. The
compactness results in the quantization of the black hole radius. Then we
quantize the two boson fields and get two abelian Kac-Moody algebras. From the
boson field, one can construct the full $W_{1+\infty}$ algebra which was used
to classify the BTZ black holes.
