Reductive subalgebras of semisimple Lie algebras and Poisson commutativity

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 $(\dim \mathfrak{g}+\mathrm{rk}\mathfrak{g})/2$.