Realization of Jordan-Kronecker invariants by Lie algebras

TL;DR

This paper characterizes which Jordan-Kronecker invariants can be realized by Lie algebras, providing complete solutions in certain cases and partial results in others, especially concerning Poisson brackets with non-constant eigenvalues.

Contribution

It completely solves the realization problem for Jordan and Kronecker cases and offers partial solutions for more complex invariants, advancing understanding of Lie algebra structures.

Findings

Any JK invariants with a 3x3 Kronecker block are possible.

Invariants with multiple 1x1 Kronecker blocks are possible.

Certain invariants with multiple maxima in Jordan tuples are impossible.

Abstract

We study what Jordan-Kronecker invariants of Lie algebras, introduced by A. V. Bolsinov and P. Zhang, are possible. We completely solve this problem in the Jordan and the Kronecker cases. We prove that any JK invariants that contain the Kronecker $3 \times 3$ block or several Kronecker $1 \times 1$ blocks are possible. For other JK invariants, with Kronecker indices $k_1, \dots, k_q$, we give a partial answer: all Jordan--Kronecker invariants with no more than $\sum_i k_i$ Jordan tuples with multiple maxima are possible; the Jordan--Kronecker invariants with more than $\sum_i k_i$ unique Jordan tuples with multiple maxima are impossible. We also desribe all JK invariants that can be realized by compatible Poisson brackets with non-constant eigenvalues