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This is the last chapter of our series of papers [1], [3], [10], [11] on transformation groups for soliton equations. In [1] a link between the KdV (Korteweg de Vries) equation and the affine Lie algebra A[^ was found: the vertex operator that affords an explicit realization of the basic representation of A{ [2] acts infinitesimally on the i functions of the KdV hierarchy. It was shown also that this link between the KdV equation and A[ comes from a similar link between the KP (Kadomtsev-Petviashvili) equation and gl(oo); the restriction to the subalgebra A{^ in gl(oo) reduces the KP hierarchy to the KdV hierarchy.

Transformation groups for soliton equations. Euclidean Lie algebras and reduction of the KP hierarchy