Reductive subalgebras of semisimple Lie algebras and Poisson commutativity

Let $\mathfrak{g}$ be a semisimple Lie algebra, $\mathfrak{h} \subset \mathfrak{g}$ a reductive subalgebra such that the orthogonal complement $\mathfrak{h}^\bot$ 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 $H \in \mathcal{S}(\mathfrak{g})^\mathfrak{g}$ taken w.r.t. $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{h}^\bot$. 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 $b(\mathfrak{g}) = \frac{1}{2} (\dim \mathfrak{g} + \mathsf{rk} \; \mathfrak{g})$.

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The first-named author is partially supported by R.F.B.R. grant no. 20-01-00515.

The second-named author is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project number 454900253.

Received 18 December 2020

Accepted 19 November 2021

Published 16 March 2023