Computing invariants and semi-invariants by means of Frobenius Lie algebras

Let be the enveloping algebra of a finite-dimensional Lie algebra over a field k of characteristic zero, its center and its semi-center. A sufficient condition is given in order for to be a polynomial algebra over k. Surprisingly, this condition holds for many Lie algebras, especially among those for which the radical is nilpotent, in which case . In particular, it allows the explicit description of for more than half of all complex, indecomposable nilpotent Lie algebras of dimension at most 7.