Casimir operators and nilpotent radicals (Q2909643)

Using the properties of algebraic Lie algebras and some results on enveloping algebras [\textit{J. Dixmier}, Algèbres enveloppantes. Paris etc.: Gauthier-Villars (1974; Zbl 0308.17007)], in this paper it is shown that Lie algebras possessing a nilpotent radical have a fundamental set of invariants that can be chosen as polynomials. The author also comments on the possibility of extending the result to those solvable non-nilpotent Lie algebras admitting polynomial invariants, whose characterization is yet unknown. In addition, it is analyzed under which conditions the number of invariants coincides with the rank of the Levi subalgebra when the latter is isomorphic to \(\mathfrak{sl}(2)\).