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

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.