Abstract
We consider cohomological and Poisson structures associated with the special tautological subbundles \(TB_{W_{1,2, \ldots n} }\) for the Birkhoff strata of the Sato Grassmannian. We show that the tangent bundles of \(TB_{W_{1,2, \ldots n} }\) are isomorphic to the linear spaces of two-coboundaries with vanishing Harrison cohomology modules. A special class of two-coboundaries is provided by a system of integrable quasilinear partial differential equations. For the big cell, it is the hierarchy of dispersionless Kadomtsev-Petvishvili (dKP) equations. We also demonstrate that the families of ideals for algebraic varieties in \(TB_{W_{1,2, \ldots n} }\) can be viewed as Poisson ideals. This observation establishes a relation between families of algebraic curves in \(TB_{W_{\hat S} }\) and coisotropic deformations of such curves of zero and nonzero genus described by hierarchies of systems of hydrodynamic type; the dKP hierarchy is such a hierarchy. We note the interrelation between cohomological and Poisson structures.
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Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 177, No. 2, pp. 231–246, November, 2013.
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Konopelchenko, B.G., Ortenzi, G. Cohomological and Poisson structures and integrable hierarchies in tautological subbundles for Birkhoff strata of the Sato Grassmannian. Theor Math Phys 177, 1479–1491 (2013). https://doi.org/10.1007/s11232-013-0117-y
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DOI: https://doi.org/10.1007/s11232-013-0117-y