Open Problems in Integrable Systems — Nijenhuis operators: singular points and global properties

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 $(M^{2n},\omega)$ be symplectic and $P$ a Poisson tensor compatible with $\omega^{-1}$, meaning every linear combination of these two tensors is Poisson. The recursion operator is $R=P^{\sharp}\circ\omega^{\flat}:TM\to TM$ and is Nijenhuis. Take $n$ independent characteristic-coefficient functions $\sigma_1,\ldots,\sigma_n$ of $R$, assumed to have independent differentials almost everywhere, and put $\Phi=(\sigma_1,\ldots,\sigma_n)$. Generically $R$ is semisimple and every eigenvalue has multiplicity two. A singular point of $R$ is one where its Jordan type is not locally constant; a singular point of $\Phi$ has $\operatorname{rank}d\Phi<n$.

A left-symmetric algebra is an algebra whose associator satisfies \[a\cdot(b\cdot c)-(a\cdot b)\cdot c=b\cdot(a\cdot c)-(b\cdot a)\cdot c.\] For a rank-$k$ critical point of an integrable system, Eliasson nondegeneracy means that the quadratic Hamiltonians induced by the critical components on the $2(n-k)$-dimensional symplectic normal space span a Cartan subalgebra of its symplectic Lie algebra.

Statement recorded in the supplied source.

Describe the singularities of $\Phi$ in terms of the singular points of $R$, their linearizations and the associated left-symmetric algebras. Find sufficient and/or necessary conditions, in these terms, for a singularity of $\Phi$ to be Eliasson nondegenerate. Retain the special generic double-eigenvalue type of $R$. [S2]

Short English statement

Relate singularities of the characteristic-coefficient integrable system to the recursion tensor and its linear algebra, and characterize when those singularities are nondegenerate.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2100514 (AMR-020-0514). 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), Problem 5.17 and its complete preceding definitions. Original question and full discussion. https://arxiv.org/abs/1804.03737.