On the coadjoint representation of $\mathbb Z_2$-contractions of reductive Lie algebras

TL;DR

This paper investigates the coadjoint representation of contractions of reductive Lie algebras derived from symmetric decompositions, proposing conjectures about their invariant properties and establishing polynomial invariants in many cases.

Contribution

It introduces a conjecture that contractions from symmetric decompositions share key properties with reductive Lie algebras, including polynomial invariants.

Findings

Proved polynomial invariants in many cases

Discussed the 'codim--2 property' and its relation to invariants

Established properties of coadjoint representations of contractions

Abstract

We study the coadjoint representation of contractions of reductive Lie algebras associated with symmetric decompositions. Let $\frak g=\frak g_0\oplus \frak g_1$ be a symmetric decomposition of a reductive Lie algebra $\frak g$. Then the semi-direct product of $\frak g_0$ and the $\frak g_0$-module $\frak g_1$ is a contraction of $\frak g$. We conjecture that these contractions have many properties in common with reductive Lie algebras. In particular, it is proved that in many cases the algebra of invariants is polynomial. We also discuss the so-called "codim--2 property" for coadjoint representations and its relationship with the structure of algebra of invariants.