Cohomological, Poisson structures and integrable hierarchies in tautological subbundles for Birkhoff strata of Sato Grassmannian

Abstract

Cohomological and Poisson structures associated with the special tautological subbundles $T{B}_{W_{1,2,\dots ,n}}$ for the Birkhoff strata of Sato Grassmannian are considered. It is shown that the tangent bundles of $T{B}_{W_{1,2,\dots ,n}}$ are isomorphic to the linear spaces of $2-$coboundaries with vanishing Harrison's cohomology modules. Special class of 2-coboundaries is provided by the systems of integrable quasilinear PDEs. For the big cell it is the dKP hierarchy. It is demonstrated also that the families of ideals for algebraic varieties in $T{B}_{W_{1,2,\dots ,n}}$ can be viewed as the Poisson ideals. This observation establishes a connection between families of algebraic curves in $T{B}_{W_{\hat{S}}}$ and coisotropic deformations of such curves of zero and nonzero genus described by hierarchies of hydrodynamical type systems like dKP hierarchy. Interrelation between cohomological and Poisson structures is noted.

15 pages, no figures, accepted in Theoretical and Mathematical Physics. arXiv admin note: text overlap with arXiv:1005.2053

References in corpus (8)

• Deformation quantization of algebraic varieties
• Coisotropic deformations of associative algebras and dispersionless integrable hierarchies
• Singular sector of the Burgers-Hopf hierarchy and deformations of hyperelliptic curves
• Dispersionless integrable equations as coisotropic deformations. Extensions and reductions
• Algebraic varieties in Birkhoff strata of the Grassmannian Gr${}^{(2)}$: Harrison cohomology and integrable systems
• Coisotropic deformations of algebraic varieties and integrable systems
• Birkhoff strata of the Grassmannian Gr${}^{(2)}$: Algebraic curves
• Birkhoff strata of Sato Grassmannian and algebraic curves