Open Problems in Integrable Systems — Argument shift method and Jordan-Kronecker invariants of Lie algebras

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $V=\mathbb R^N$. A quadratic Poisson bracket is given by homogeneous quadratic polynomials $P_{ij}(x)=-P_{ji}(x)$ such that \[\{f,g\}=\sum_{i,j=1}^N P_{ij}(x)\frac{\partial f}{\partial x_i}\frac{\partial g}{\partial x_j}\] satisfies the Jacobi identity. Let $2r$ be the maximum rank of the skew-symmetric matrix $P(x)$. A complete polynomial integrable system is a collection of $N-r$ polynomial functions with pairwise vanishing brackets and with linearly independent differentials almost everywhere on $V$.

Statement recorded in the supplied source.

Does every quadratic Poisson bracket on a finite-dimensional vector space admit a complete polynomial integrable system in this sense? [S2]

Short English statement

Does every quadratic Poisson structure on a vector space have enough independent commuting polynomial functions for Liouville integrability?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2100507 (AMR-020-0507). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Alexey Bolsinov, Vladimir S. Matveev, Eva Miranda and Serge Tabachnikov, Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), Question 5.9 and its complete preceding definitions. Original question and full discussion. https://arxiv.org/abs/1804.03737.