Cohomological and Poisson structures and integrable hierarchies in tautological subbundles for Birkhoff strata of the Sato Grassmannian
type: Item
instance of: scholarly article
title: Cohomological and Poisson structures and integrable hierarchies in tautological subbundles for Birkhoff strata of the Sato Grassmannian
zbMATH Open document ID: 1301.37042
author: B. Konopelchenko
author: G. Ortenzi
published in: Theoretical and Mathematical Physics
publication date: 2014-10-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1306.4571
review text: In this article, the authors consider cohomological and Poisson structures associated with the special tautological subbundles \(TB_{W_{1,2,\dots,n}}\) for the Birkhoff strata of the Sato Grassmannian. It is shown that the tangent bundles of \(TB_{W_{1,2,\dots,n}}\) 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-Petviashvili (dKP) equations. It is also demonstrated that the families of ideals for algebraic varieties in \(TB_{W_{1,2,\dots,n}}\) 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. As a conclusion, the interrelation between cohomological and Poisson structures is noted.
reviewed by: Vladislav Nikolaevich Dumachev
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 70G45
zbMATH DE Number: 6353914
zbMATH Keywords: Birkhoff stratum
zbMATH Keywords: Harrison cohomology
zbMATH Keywords: integrable system
MaRDI profile type: Publication
OpenAlex ID: W3103359511
arXiv ID: 1306.4571
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DOI: 10.1007/S11232-013-0117-Y
description: scientific article; zbMATH DE number 6353914