On the coadjoint representation of ${\mathbb{Z}}_2$-contractions of reductive Lie algebras

Abstract

We study the coadjoint representation of contractions of reductive Lie algebras associated with symmetric decompositions. Let $\mathfrak{g}={\mathfrak{g}}_0\oplus {\mathfrak{g}}_1$ be a symmetric decomposition of a reductive Lie algebra $\mathfrak{g}$. Then the semi-direct product of ${\mathfrak{g}}_0$ and the ${\mathfrak{g}}_0$-module ${\mathfrak{g}}_1$ is a contraction of $\mathfrak{g}$. 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.

25 pages, 3 tables