Reductive subalgebras of semisimple Lie algebras and Poisson commutativity

TL;DR

This paper investigates conditions under which certain subalgebras related to reductive subalgebras of semisimple Lie algebras are Poisson commutative, correcting a previous claim and exploring specific cases like abelian and Cartan subalgebras.

Contribution

It provides a counterexample to a 1983 claim about Poisson commutativity and offers a criterion for when these subalgebras are Poisson commutative, especially for abelian and Cartan subalgebras.

Findings

Counterexample to Bogoyavlenski's claim from 1983.

Criterion for Poisson commutativity of subalgebras ${\\mathcal Z}$.

${\mathcal Z}$ is polynomial with maximal transcendence degree when ${\mathfrak h}$ is a Cartan subalgebra.

Abstract

Let $\mathfrak g$ be a semisimple Lie algebra, $\mathfrak h\subset\mathfrak g$ a reductive subalgebra such that $\mathfrak h^\perp$ is a complementary $\mathfrak h$-submodule of $\mathfrak g$. In 1983, Bogoyavlenski claimed that one obtains a Poisson commutative subalgebra of the symmetric algebra ${\mathcal S}(\mathfrak g)$ by taking the subalgebra ${\mathcal Z}$ generated by the bi-homogeneous components of all $H\in{\mathcal S}(\mathfrak g)^{\mathfrak g}$. But this is false, and we present a counterexample. We also provide a criterion for the Poisson commutativity of such subalgebras ${\mathcal Z}$. As a by-product, we prove that ${\mathcal Z}$ is Poisson commutative if $\mathfrak h$ is abelian and describe ${\mathcal Z}$ in the special case when $\mathfrak h$ is a Cartan subalgebra. In this case, ${\mathcal Z}$ appears to be polynomial and has the maximal transcendence degree $(\mathrm{dim}\,\mathfrak g+\mathrm{rk}\,\mathfrak g)/2$.