Sine-Gordon Equation and Representations of Affine Kac-Moody Algebra \hat{sl_2}

The goal of this paper is to give a representation-theoretic interpretation of the sine-Gordon equation. We consider a vertex operator representation of affine Kac-Moody algebra \hat{sl_2} on the space of differential operators. In this formulation, the tau-function becomes a function of non-commuting variables. Using the skew Casimir operators, we obtain a hierarchy of equations in Hirota form that contains sine-Gordon, KdV and mKdV equations and construct their soliton solutions.