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

TL;DR

This paper classifies two- and three-component hydrodynamic Poisson structures with degenerate metrics and investigates their deformations, revealing non-trivial deformations dependent on arbitrary functions, unlike the trivial case with non-degenerate metrics.

Contribution

It provides a complete classification of degenerate metric Poisson structures and analyzes their non-trivial deformations, showing the non-vanishing of the second Poisson-Lichnerowicz cohomology group.

Findings

Degenerate Poisson structures admit non-trivial deformations.

Deformations depend on arbitrary functions, unlike the trivial non-degenerate case.

Second Poisson-Lichnerowicz cohomology group does not vanish in the degenerate case.

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.