On deformations of one-dimensional Poisson structures of hydrodynamic type with degenerate metric

Abstract

We provide a complete list of two- and three-component Poisson structures of hydrodynamic type with degenerate metric, and study their homogeneous deformations. In the non-degenerate case any such deformation is trivial, that is, can be obtained via Miura transformation. We demonstrate that in the degenerate case this class of deformations is non-trivial, and depends on a certain number of arbitrary functions. This shows that the second Poisson-Lichnerowicz cohomology group does not vanish.

49 pages; typos and Theorem 1 corrected; revised argument in Section 3.1; Section 3.2 added