  We investigate the $W_{\infty}$ algebra in the integer quantum Hall effects.
Defining the simplest vacuum, the Dirac sea, we evaluate the central extension
for this algebra. A new algebra which contains the central extension is called
the $W_{1+\infty}$ algebra. We show that this $W_{1+\infty}$ algebra is an
origin of the Kac-Moody algebra which determines the behavior of edge states of
the system. We discuss the relation between the $W_{1+\infty}$ algebra and the
incompressibility of the integer quantum Hall system.
