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

TL;DR

This paper generalizes the construction of commutative subalgebras in polynomial Poisson algebras associated with Lie algebras, extending Mischenko-Fomenko subalgebras by incorporating shifts of all semi-invariants, and proves they maintain the same transcendence degree.

Contribution

It introduces a new class of commutative subalgebras in symmetric algebras of Lie algebras by extending existing constructions with semi-invariant shifts, preserving their transcendence degree.

Findings

Extended Mischenko-Fomenko subalgebras include all semi-invariant shifts.

These subalgebras have the same transcendence degree as the original extended subalgebras.

The generalization broadens the understanding of polynomial Poisson algebra structures.

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 $\tilde{\mathcal{F}}_a$.