Geodesic
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 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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},
     year = {2013},
     volume = {177},
     number = {2},
     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
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
%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/

[1] B. G. Konopelchenko, G. Ortenzi, Birkhoff strata of Sato Grassmannian and algebraic curves, arXiv: 1005.2053 | MR

[2] B. G. Konopelchenko, G. Ortenzi, J. Phys. A, 44:46 (2011), 465201, 31 pp., arXiv: 1102.0700 | DOI | MR | Zbl

[3] B. G. Konopelchenko, Dzh. Ortentsi, TMF, 167:3 (2011), 448–464 | DOI | DOI

[4] E. Pressli, G. Sigal, Gruppy petel, Mir, M., 1990 | MR | Zbl

[5] G. Segal, G. Wilson, Inst. Hautes Études Sci. Publ. Math., 61 (1985), 5–65 | DOI | MR | Zbl

[6] V. Khodzh, D. Pido, Metody algebraicheskoi geometrii, v. 1, IL, M., 1954 | MR

[7] I. R. Shafarevich, Osnovy algebraicheskoi geometrii, Nauka, M., 1988 | MR | Zbl

[8] F. Griffits, Dzh. Kharris, Printsipy algebraicheskoi geometrii, Mir, M., 1982 | MR | MR | Zbl

[9] Dzh. Kharris, Algebraicheskaya geometriya. Nachalnyi kurs, MTsNMO, M., 2005 | MR | Zbl

[10] V. E. Zakharov, Funkts. analiz i ego pril., 14:2 (1980), 15–24 | DOI | MR | Zbl

[11] I. M. Krichever, Funkts. analiz i ego pril., 22:3 (1988), 37–52 | DOI | MR | Zbl

[12] Y. Kodama, Phys. Lett. A, 129:4 (1988), 223–226 | DOI | MR

[13] K. Takasaki, T. Takebe, Internat. J. Modern Phys. A, 7, suppl. 1B (1992), 889–922, arXiv: hep-th/9112046 | DOI | Zbl

[14] I. M. Krichever, Comm. Pure Appl. Math., 47:4 (1994), 437–475 | DOI | MR | Zbl

[15] B. G. Konopelchenko, F. Magri, Commun. Math. Phys., 274:3 (2007), 627–658, arXiv: nlin/0606069 | DOI | MR | Zbl

[16] G. Hochschild, Ann. Math. (2), 46:1 (1945), 58–67 | DOI | MR | Zbl

[17] D. K. Harrison, Trans. Amer. Math. Soc., 104 (1962), 191–204 | DOI | MR | Zbl

[18] M. Gerstenhaber, Ann. Math. (2), 79:1 (1964), 59–103 | DOI | MR | Zbl

[19] A. Nijenhuis, R. W. Richardson Jr., J. Algebra, 9:1 (1968), 42–53 | DOI | MR | Zbl

[20] M. Barr, J. Algebra, 8:3 (1968), 314–323 | DOI | MR | Zbl

[21] V. P. Palamodov, UMN, 31:3(189) (1976), 129–194 | DOI | MR | Zbl

[22] M. Schlessinger, J. Stasheff, J. Pure Appl. Algebra, 38:2–3 (1985), 313–322 | DOI | MR | Zbl

[23] S. Gutt, Lett. Math. Phys., 39:2 (1997), 157–162 | DOI | MR | Zbl

[24] M. Kontsevich, Lett. Math. Phys., 56:3 (2001), 271–294 | DOI | MR | Zbl

[25] C. Frønsdal, “Harrison cohomology and abelian deformation quantization on algebraic varieties”, Deformation Quantization (Strasbourg, May 31 – June 2, 2001), v. 1, IRMA Lectures in Mathematics and Theoretical Physics, ed. G. Halbout, de Gruyter, Berlin, 2002, 149–161 | MR

[26] Y. Kodama, B. G. Konopelchenko, J. Phys. A, 35:31 (2002), L489–L500 | DOI | MR | Zbl

[27] A. Weinstein, J. Math. Soc. Japan, 40:4 (1988), 705–727 | DOI | MR | Zbl

[28] A. Givental, K. Bumsig, Commun. Math. Phys., 168:3 (1995), 609–641, arXiv: hep-th/9312096 | DOI | MR | Zbl

[29] B. G. Konopelchenko, G. Ortenzi, J. Phys. A, 42:41 (2009), 415207, 18 pp. | DOI | MR | Zbl

[30] B. G. Konopelchenko, F. Magri, TMF, 151:3 (2007), 439–457 | DOI | DOI | MR | Zbl