Central invariants revisited
Guido Carlet; Reinier Kramer; Sergey Shadrin
Journal de l'École polytechnique — Mathématiques, Volume 5 (2018), p. 149-175
We use refined spectral sequence arguments to calculate known and previously unknown bi-Hamiltonian cohomology groups, which govern the deformation theory of semi-simple bi-Hamiltonian pencils of hydrodynamic type with one independent and N dependent variables. In particular, we rederive the result of Dubrovin-Liu-Zhang that these deformations are parametrized by the so-called central invariants, which are N smooth functions of one variable.
Nous utilisons des arguments raffinés de suite spectrale pour calculer des groupes de cohomologie bihamiltonienne, certains déjà connus et d’autres non, qui gouvernent la théorie des déformations de pinceaux bihamiltoniens semi-simples de type hydrodynamique avec une variable indépendante et N variables dépendantes. En particulier, nous retrouvons le résultat de Dubrovin-Liu-Zhang disant que ces déformations sont paramétrées par les invariants centraux, qui sont N fonctions lisses d’une variable.
Received : 2016-12-20
Accepted : 2017-11-23
Published online : 2017-12-12
DOI : https://doi.org/10.5802/jep.66
Classification:  37K10,  53D17,  58A20
Keywords: Poisson structures of hydrodynamic type, deformations of bi-Hamiltonian structures, bi-Hamiltonian cohomology, central invariants
@article{JEP_2018__5__149_0,
     author = {Guido Carlet and Reinier Kramer and Sergey Shadrin},
     title = {Central invariants revisited},
     journal = {Journal de l'\'Ecole polytechnique --- Math\'ematiques},
     publisher = {\'Ecole polytechnique},
     volume = {5},
     year = {2018},
     pages = {149-175},
     doi = {10.5802/jep.66},
     zbl = {06988576},
     mrnumber = {3738511},
     language = {en},
     url = {https://jep.centre-mersenne.org/item/JEP_2018__5__149_0}
}
Carlet, Guido; Kramer, Reinier; Shadrin, Sergey. Central invariants revisited. Journal de l'École polytechnique — Mathématiques, Volume 5 (2018) pp. 149-175. doi : 10.5802/jep.66. https://jep.centre-mersenne.org/item/JEP_2018__5__149_0/

[Bar08] A. Barakat - “On the moduli space of deformations of bihamiltonian hierarchies of hydrodynamic type”, Adv. Math. 219 (2008) no. 2, p. 604-632 | Article | MR 2435651

[CPS15] G. Carlet, H. Posthuma & S. Shadrin - “Deformations of semisimple Poisson pencils of hydrodynamic type are unobstructed” (2015), arXiv:1501.04295 | Zbl 1387.53105

[CPS16a] G. Carlet, H. Posthuma & S. Shadrin - “Bihamiltonian cohomology of KdV brackets”, Comm. Math. Phys. 341 (2016) no. 3, p. 805-819 | Article | MR 3452272 | Zbl 1362.37124

[CPS16b] G. Carlet, H. Posthuma & S. Shadrin - “The bi-Hamiltonian cohomology of a scalar Poisson pencil”, Bull. London Math. Soc. 48 (2016) no. 4, p. 617-627 | Article | MR 3532137 | Zbl 1350.53106

[DLZ06] B. Dubrovin, S.-Q. Liu & Y. Zhang - “On Hamiltonian perturbations of hyperbolic systems of conservation laws. I. Quasi-triviality of bi-Hamiltonian perturbations”, Comm. Pure Appl. Math. 59 (2006) no. 4, p. 559-615 | Article | MR 2199786 | Zbl 1108.35112

[DMS05] L. Degiovanni, F. Magri & V. Sciacca - “On deformation of Poisson manifolds of hydrodynamic type”, Comm. Math. Phys. 253 (2005) no. 1, p. 1-24 | MR 2105635 | Zbl 1108.53044

[DN83] B. Dubrovin & S. Novikov - “Hamiltonian formalism of one-dimensional systems of the hydrodynamic type and the Bogolyubov-Whitham averaging method”, Dokl. Akad. Nauk SSSR 270 (1983) no. 4, p. 781-785 | MR 715332 | Zbl 0553.35011

[DVLS16] A. Della Vedova, P. Lorenzoni & A. Savoldi - “Deformations of non-semisimple Poisson pencils of hydrodynamic type”, Nonlinearity 29 (2016) no. 9, p. 2715-2754 | Article | MR 3544805 | Zbl 1348.37099

[DZ01] B. Dubrovin & Y. Zhang - “Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants” (2001), arXiv:math/0108160

[Fer01] E. V. Ferapontov - “Compatible Poisson brackets of hydrodynamic type”, J. Phys. A 34 (2001), p. 2377-2388 | Article | MR 1831303 | Zbl 1010.37044

[Get02] E. Getzler - “A Darboux theorem for Hamiltonian operators in the formal calculus of variations”, Duke Math. J. 111 (2002) no. 3, p. 535-560 | Article | MR 1885831 | Zbl 1100.32008

[Lor02] P. Lorenzoni - “Deformations of bihamiltonian structures of hydrodynamic type”, J. Geom. Phys. 44 (2002) no. 2-3, p. 331-375 | Article | MR 1969787

[LZ05] S.-Q. Liu & Y. Zhang - “Deformations of semisimple bihamiltonian structures of hydrodynamic type”, J. Geom. Phys. 54 (2005) no. 4, p. 427-453 | Article | MR 2144711 | Zbl 1079.37058

[LZ13] S.-Q. Liu & Y. Zhang - “Bihamiltonian cohomologies and integrable hierarchies I: A special case”, Comm. Math. Phys. 324 (2013) no. 3, p. 897-935 | MR 3123540 | Zbl 1318.37019

[Mag78] F. Magri - “A simple model of the integrable Hamiltonian equation”, J. Math. Phys. 19 (1978) no. 5, p. 1156-1162 | Article | MR 488516

[Zha02] Y. Zhang - “Deformations of the bihamiltonian structures on the loop space of Frobenius manifolds”, J. Nonlinear Math. Phys. 9 (2002) no. sup1, p. 243-257 | Article | MR 1900199 | Zbl 1362.37138