Cohomological and Poisson structures and integrable hierarchies in tautological subbundles for Birkhoff strata of the~Sato Grassmannian
Teoretičeskaâ i matematičeskaâ fizika, Tome 177 (2013) no. 2, pp. 231-246
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We 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. We show 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–Petvishvili (dKP) equations. We also demonstrate 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 $TB_{W_{\widehat S}}$ 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.
Keywords:
Birkhoff stratum, Harrison cohomology, integrable system.
@article{TMF_2013_177_2_a2,
author = {B. G. Konopelchenko and G. Ortenzi},
title = {Cohomological and {Poisson} structures and integrable hierarchies in tautological subbundles for {Birkhoff} strata of {the~Sato} {Grassmannian}},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {231--246},
publisher = {mathdoc},
volume = {177},
number = {2},
year = {2013},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2013_177_2_a2/}
}
TY - JOUR AU - B. G. Konopelchenko AU - G. Ortenzi TI - Cohomological and Poisson structures and integrable hierarchies in tautological subbundles for Birkhoff strata of the~Sato Grassmannian JO - Teoretičeskaâ i matematičeskaâ fizika PY - 2013 SP - 231 EP - 246 VL - 177 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TMF_2013_177_2_a2/ LA - ru ID - TMF_2013_177_2_a2 ER -
%0 Journal Article %A B. G. Konopelchenko %A G. Ortenzi %T Cohomological and Poisson structures and integrable hierarchies in tautological subbundles for Birkhoff strata of the~Sato Grassmannian %J Teoretičeskaâ i matematičeskaâ fizika %D 2013 %P 231-246 %V 177 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/item/TMF_2013_177_2_a2/ %G ru %F TMF_2013_177_2_a2
B. G. Konopelchenko; G. Ortenzi. Cohomological and Poisson structures and integrable hierarchies in tautological subbundles for Birkhoff strata of the~Sato Grassmannian. Teoretičeskaâ i matematičeskaâ fizika, Tome 177 (2013) no. 2, pp. 231-246. http://geodesic.mathdoc.fr/item/TMF_2013_177_2_a2/