Some semi-direct products with free algebras of symmetric invariants

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 $\mathfrak{q}=\mathfrak{g}⋉V$, 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 symmetries invariants of $\mathfrak{q}$ and describe all semi-direct products related to the defining representation of ${\mathfrak{s}\mathfrak{l}}_n$ with $\mathbb C[\mathfrak q^*]^{\mathfrak q}$ being a free algebra.