Given a semidirect product $\frak{ g}=\frak{ s}\uplus\frak{ r}$ of semisimple Lie algebras $\frak{ s}$ and solvable algebras $\frak{ r}$, we construct polynomial operators in the enveloping algebra $\mathcal{ U}(\frak{ g})$ of $\frak{ g}$ that commute with $\frak{ r}$ and transform like the generators of $\frak{ s}$, up to a functional factor that turns out to be a Casimir operator of $\frak{ r}$. Such operators are said to generate a virtual copy of $\frak{ s}$ in $\mathcal{ U}(\frak{ g})$, and allow to compute the Casimir operators of $\frak{ g}$ in closed form, using the classical formulae for the invariants of $\frak{ s}$. 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.
Categories: math-ph math.MP