The Grassmannian and the Lax pair approaches to the Kadomtsev-Petviashvili (KP) hierarchy are analyzed in the framework of (formal) principal fiber bundles. The underlying factorization problem is formulated as a local triviality condition. In particular the common object in the two approaches — the Baker function — coordinatizes the base space, the fiber consists of certain generalized differential operators. An example is constructed to show that despite the apparent similarity to the splitting in , the local triviality condition cannot be given a group theoretic interpretation.
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Josef Dorfmeister, Jacek Szmigielski, Principal Fiber Bundle Interpretation of the KP-Hierarchy. Publ. Res. Inst. Math. Sci. 28 (1992), no. 4, pp. 503–533DOI 10.2977/PRIMS/1195168205