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.

For a finite-dimensional complex Lie algebra $\mathfrak g$ and $x,a\in\mathfrak g^*$, form the alternating pencil \[\mathcal A_{x+\lambda a}(u,v)=\langle x+\lambda a,[u,v]\rangle.\] Its Jordan--Kronecker decomposition is the simultaneous congruence canonical form of its pair of alternating forms: even-dimensional Jordan blocks at characteristic parameters and odd-dimensional Kronecker blocks. Its algebraic type records the number and sizes of Kronecker blocks and the sizes of Jordan blocks grouped by equal characteristic parameter; the numerical characteristic values themselves are ignored. For generic $(x,a)$ this type is the Jordan--Kronecker invariant of $\mathfrak g$.

Statement recorded in the supplied source.

Characterize the algebraic types realizable by such generic argument-shift pencils as $\mathfrak g$ ranges over finite-dimensional complex Lie algebras. Determine all restrictions on the combinations of Jordan and Kronecker blocks and which admissible types can actually occur. [S2]

Short English statement

Which simultaneous alternating-form canonical types occur as generic argument-shift pencils of Lie algebras?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2100505 (AMR-020-0505). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 A. Bolsinov, V. S. Matveev, E. Miranda and S. Tabachnikov, Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), arXiv:1804.03737, Question 5.7 and generic Jordan-Kronecker definition. https://arxiv.org/abs/1804.03737.