Sine-Gordon equation and representations of affine Kac-Moody algebra sl2

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.

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