New techniques for calculation of Jordan-Kronecker invariants for Lie algebras and Lie algebra representations

TL;DR

This paper presents two new methods to simplify the calculation of Jordan-Kronecker invariants for Lie algebras and their representations, enhancing understanding of their structure and relationships.

Contribution

The paper introduces stratification techniques and dual representation analysis to streamline the computation of Jordan-Kronecker invariants in Lie algebra theory.

Findings

Stratification restricts possible invariants.

Invariants of semi-direct sums relate to dual representations.

Methods simplify complex calculations.

Abstract

We introduce two novel techniques that simplify calculation of Jordan-Kronecker invariants for a Lie algebra $\mathfrak{g}$ and for a Lie algebra representation $\rho$. First, the stratification of matrix pencils under strict equivalence puts restrictions on the Jordan-Kronecker invariants. Second, we show that the Jordan-Kronecker invariants of a semi-direct sum $\mathfrak{g} \ltimes_{\rho} V$ are sometimes determined by the Jordan-Kronecker invariants of the dual Lie algebra representation $\rho^*$.