Shifts of semi-invariants and complete commutative subalgebras in polynomial Poisson algebras

Abstract

We study commutative subalgebras in the symmetric algebra $S(\mathfrak{g})$ of a finite-dimensional Lie algebra $\mathfrak{g}$. A. M. Izosimov introduced extended Mischenko-Fomenko subalgebras $\tilde{\mathcal{F}}_a$ and gave a completeness criterion for them. We generalize his construction and extend Mischenko-Fomenko subalgebras with the shifts of all semi-invariants of $\mathfrak{g}$. We prove that the new commutative subalgebras have the same transcendence degree as $\tilde{\mathcal{F}}_a$.