W Algebra in the Integer Quantum Hall Effects

We investigate the W∞ 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 W1+∞ algebra. The W1+∞ algebra is considered to play an important role for the incompressibility which is a property of the bulk of the electron liquid. We show that the W1+∞ algebra is crucial for the edge states and it is an origin of the Kac-Moody algebra which determines the behavior of edge states of the system.