Some semi-direct products with free algebras of symmetric invariants

Abstract: Let $\mathfrak g$ be a complex reductive Lie algebra and $V$ the underling 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 $\mathbb C[\mathfrak q^*]^{\mathfrak q}$ of symmetries invariants of $\mathfrak q$ and describe all semi-direct products related to the defining representation of $\mathfrak{sl}_n$ with $\mathbb C[\mathfrak q^*]^{\mathfrak q}$ being a free algebra.