Virtual copies of semisimple Lie algebras in enveloping algebras of semidirect products and Casimir operators

Given a semidirect product $\mathfrak{g}=\mathfrak{s}\uplus \mathfrak{r}$ of semisimple Lie algebras $\mathfrak{s}$ and solvable algebras $\mathfrak{r}$, we construct polynomial operators in the enveloping algebra $\mathcal{U}(\mathfrak{g})$ of $\mathfrak{g}$ that commute with $\mathfrak{r}$ and transform like the generators of $\mathfrak{s}$, up to a functional factor that turns out to be a Casimir operator of $\mathfrak{r}$. Such operators are said to generate a virtual copy of $\mathfrak{s}$ in $\mathcal{U}(\mathfrak{g})$, and allow to compute the Casimir operators of $\mathfrak{g}$ in closed form, using the classical formulae for the invariants of $\mathfrak{s}$. The behavior of virtual copies with respect to contractions of Lie algebras is analyzed. Applications to the class of Hamilton algebras and their inhomogeneous extensions are given.