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Let $\mathfrak g$ be a complex reductive Lie algebra and $V$ the underlingvector space of a finite-dimensional representation of $\mathfrak g$. Then onecan consider a new Lie algebra $\mathfrak q=\mathfrak g{\ltimes} V$, which is asemi-direct product of $\mathfrak g$ and an Abelian ideal $V$. We outlineseveral results on the algebra $\mathbb C[\mathfrak q^*]^{\mathfrak q}$ ofsymmetries invariants of $\mathfrak q$ and describe all semi-direct productsrelated to the defining representation of $\mathfrak{sl}_n$ with $\mathbbC[\mathfrak q^*]^{\mathfrak q}$ being a free algebra.

Some semi-direct products with free algebras of symmetric invariants