Sine-Gordon Equation and Representations of Affine Kac–Moody Algebra sl2̂

We give a representation-theoretic interpretation of the sine-Gordon equation using the action of affine Kac–Moody algebra sl2 on the Weyl algebra. In this setup τ-functions become functions in non-commuting variables. The skew Casimir operators that we introduce here, give rise to a hierarchy of partial differential equations that includes the sine-Gordon equation and two copies of the Korteweg–de Vries equation.