A degree inequality for Lie algebras with a regular Poisson semi-center (Q2268816)

Let \({\mathfrak g}\) be a finite dimensional Lie algebra over an algebraically closed field of characteristic zero. Denote by \((S{\mathfrak g})^{\mathfrak g}_{si}\) the Poisson semi-center of \(S{\mathfrak g}\) and by \(i({\mathfrak g})\) the index of \({\mathfrak g}\). In the main result the authors prove that if \((S{\mathfrak g})^{\mathfrak g}_{si}\) is freely generated by homogeneous elements \(f_1,\dots, f_r\), then \(\sum_{i=1}^r \deg f_i \leq \frac{1}{2}\left(\dim {\mathfrak g}+i({\mathfrak g})\right)\). Such a result was previously known in many special cases. In particular, it has been considered for semisimple Lie algebras by Chevalley (see Theorem 7.3.8 in [\textit{J. Dixmier}, Enveloping algebras. Graduate Studies in Mathematics. 11. Providence, RI: American Mathematical Society (AMS) (1996; Zbl 0867.17001)]), for Frobenius Lie algebras by \textit{L. Delvaux, E. Nauwelaerts} and \textit{A. I. Ooms} in [J. Algebra 94, 324--346 (1985; Zbl 0594.17010)], and for Borel subalgebras of semisimple Lie algebras by \textit{A. Joseph} in [J. Algebra 48, 241--289 (1977; Zbl 0405.17007)].