Deformations of non semisimple Poisson pencils of hydrodynamic type

We study deformations of two-component non semisimple Poisson pencils of hydrodynamic type associated with Balinski\v{\i}-Novikov algebras. We show that in most cases the second order deformations are parametrized by two functions of a single variable. It turns out that one function is invariant with respect to the subgroup of Miura transformations preserving the dispersionless limit and another function is related to a one-parameter family of truncated structures. In two expectional cases the second order deformations are parametrized by four functions. Among them two are invariants and two are related to a two-parameter family of truncated structures. We also study the lift of deformations of n-component semisimple structures. This example suggests that deformations of non semisimple pencils corresponding to the lifted invariant parameters are unobstructed.

💡 Research Summary

The paper investigates deformations of two‑component non‑semisimple Poisson pencils of hydrodynamic type that arise from two‑dimensional Balinski‑Novikov algebras together with invariant symmetric bilinear forms. In the semisimple setting, Dubrovin‑Zhang theory tells us that deformations are classified by central invariants—functions of the canonical coordinates that remain unchanged under Miura transformations preserving the dispersionless limit. The non‑semisimple case, however, lacks canonical coordinates and a normal‑form theorem, making a systematic study more difficult.

The authors focus on the five families of two‑dimensional Balinski‑Novikov algebras listed in Bai‑Meng’s classification: T3, N3, N4 (with η₁₁ = 0), N5, and N6 (with a parameter κ). For each algebra they write the pair of compatible flat metrics (g¹, g²) and the constant metric η, and compute the affine matrix L = g² η⁻¹. The Poisson pencil is then ω² − λ ω¹, where ω¹ = η ∂ₓ and ω² = (g² ∂ₓ + b uₓ) with linear dependence on the fields u¹, u².

A k‑th order deformation of the pencil is a formal series Π_λ = ω² − λ ω¹ + ∑_{k≥1} ε^{k} Π^{(k)}_λ, with the requirement that the Schouten bracket