Some Semi-Direct Products with Free Algebras of Symmetric Invariants

Abstract

Let $\mathfrak{g}$ be a complex reductive Lie algebra and V the underlying vector space of a finite-dimensional representation of $\mathfrak{g}$. Then one can consider a new Lie algebra $\mathfrak{q} = \mathfrak{g}\ltimes V$, which is a semi-direct product of $\mathfrak{g}$ and an Abelian ideal V. We outline several results on the algebra $\mathbb{C}[\mathfrak{q}^{{\ast}}]^{\mathfrak{q}}$ of symmetric invariants of $\mathfrak{q}$ and describe all semi-direct products related to the defining representation of $\mathfrak{s}\mathfrak{l}_{n}$ with $\mathbb{C}[\mathfrak{q}^{{\ast}}]^{\mathfrak{q}}$ being a free algebra.