Poisson cohomology of scalar multidimensional Dubrovin–Novikov brackets

Guido Carlet (1) , Matteo Casati (2) , Sergey Shadrin (1)

1 Korteweg-de Vries Institute for Mathematics
2 SISSA / ISAS - Scuola Internazionale Superiore di Studi Avanzati / International School for Advanced Studies

We compute the Poisson cohomology of a scalar Poisson bracket of Dubrovin–Novikov type with $D$ independent variables. We find that the second and third cohomology groups are generically non-vanishing in $D > 1$ . Hence, in contrast with the $D = 1$ case, the deformation theory in the multivariable case is non-trivial.

Mots clés: Poisson brackets of hydrodynamic type; Poisson cohomology; Deformations of Poisson structures

Domaines: Mathématiques [math]

Guido Carlet, Matteo Casati, Sergey Shadrin. Poisson cohomology of scalar multidimensional Dubrovin–Novikov brackets. Journal of Geometry and Physics, 2017, 114, pp.404 - 419. ⟨10.1016/j.geomphys.2016.12.008⟩. ⟨hal-01660609⟩