Poisson-Commutative Subalgebras of 𝒮(𝔤) associated with Involutions

The symmetric algebra ${\mathcal{S}}({{\mathfrak{g}}})$ of a reductive Lie algebra ${{\mathfrak{g}}}$ is equipped with the standard Poisson structure, that is, the Lie–Poisson bracket. Poisson-commutative subalgebras of ${\mathcal{S}}({{\mathfrak{g}}})$ attract a great deal of attention because of their relationship to integrable systems and, more recently, to geometric representation theory. The transcendence degree of a Poisson-commutative subalgebra ${\mathcal C}\subset{\mathcal{S}}({{\mathfrak{g}}})$ is bounded by the “magic number” ${\boldsymbol{b}}({{\mathfrak{g}}})$ of ${{\mathfrak{g}}}$. There are two classical constructions of $\mathcal C$ with ${\textrm{tr.deg}}\,{\mathcal C}={\boldsymbol{b}}({{\mathfrak{g}}})$. The 1st one is applicable to $\mathfrak{gl}_n$ and $\mathfrak{so}_n$ and uses the Gelfand–Tsetlin chains of subalgebras. The 2nd one is known as the “argument shift method” of Mishchenko–Fomenko. We generalise the Gelfand–Tsetlin approach to chains of almost arbitrary symmetric subalgebras. Our method works for all types. Starting from a symmetric decompositions ${{\mathfrak{g}}}={{\mathfrak{g}}}_0\oplus{{\mathfrak{g}}}_1$, Poisson-commutative subalgebras ${{\mathcal{Z}}},\tilde{{\mathcal{Z}}}\subset{\mathcal{S}}({{\mathfrak{g}}})^{{{\mathfrak{g}}}_0}$ of the maximal possible transcendence degree are constructed. If the ${{\mathbb{Z}}}_2$-contraction ${{\mathfrak{g}}}_0\ltimes{{\mathfrak{g}}}_1^{\textsf{ab}}$ has a polynomial ring of symmetric invariants, then $\tilde{{\mathcal{Z}}}$ is a polynomial maximal Poisson-commutative subalgebra of ${\mathcal{S}}({{\mathfrak{g}}})^{{{\mathfrak{g}}}_0}$ and its free generators are explicitly described.