Central invariants revisited

TL;DR

This paper employs advanced spectral sequence techniques to compute bi-Hamiltonian cohomology groups, reaffirming that deformations of semi-simple bi-Hamiltonian structures are characterized by central invariants, which are smooth functions of one variable.

Contribution

It introduces refined spectral sequence methods to calculate bi-Hamiltonian cohomology, including previously unknown groups, and rederives the role of central invariants in deformation theory.

Findings

Computed bi-Hamiltonian cohomology groups using spectral sequences.

Reaffirmed that deformations are parametrized by central invariants.

Identified new cohomology groups relevant to deformation theory.

Abstract

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.