Jordan-Kronecker invariants of Lie algebra representations: examples and computations

TL;DR

This paper computes Jordan-Kronecker invariants for various Lie algebra representations, providing explicit examples and calculations for classical Lie algebras and their actions on forms.

Contribution

It offers new explicit computations of Jordan-Kronecker invariants for multiple Lie algebra representations, expanding understanding of their structure.

Findings

Computed invariants for sums of standard representations of classical Lie algebras.

Analyzed invariants for the Lie algebra of upper triangular matrices.

Determined invariants for the differential of congruence actions on forms.

Abstract

In these paper we compute Jordan-Kronecker invariants of Lie algebra representations, introduced earlier by A.V. Bolsinov, A.M. Izosimov and I.K. Kozlov, for a number of representations. In particular, we compute them for the sums of standard representations of $\operatorname{gl}(n)$, $\operatorname{sl}(n)$, $\operatorname{so}(n)$, $\operatorname{sp}(n)$, and the Lie algebra of upper triangular matrices $\operatorname{b}(n)$; the standard representation of Lie algebra of strictly upper triangular matrices $\operatorname{n}(n)$; and for the differential of the congruence action of $\operatorname{GL}(n)$ and $\operatorname{SL}(n)$ on symmetric forms and skew-symmetric forms.