A note on Jordan-Kronecker invariants of semi-direct sums of sl(n) with a commutative ideal

TL;DR

This paper investigates the Jordan-Kronecker invariants of semi-direct sums of sl(n) with a commutative ideal, focusing on cases where n is congruent to ±1 modulo k, extending previous classifications.

Contribution

It extends the classification of Jordan-Kronecker invariants to new cases where n ≡ ±1 mod k, complementing prior work on other parameter regimes.

Findings

Jordan-Kronecker invariants characterized for n ≡ ±1 mod k

Extension of classification beyond previous k > n or n multiple of k cases

Provides a complete description in these new cases

Abstract

K. S. Vorushilov described Jordan-Kronecker invariants for semi-direct sums $\operatorname{sl} \ltimes \left(\mathbb{C}^n\right)^k$ if $k > n$ or if $n$ is a multiple of $k$. We describe the Jordan-Kronecker invariants in the cases $n \equiv \pm 1 \pmod k$.