Abstract
Abstract The symmetric algebra ${\mathcal{S}}({{\mathfrak{g}}})$ of a reductive Lie algebra ${{\mathfrak{g}}}$ is equipped with the standard Poisson structure, that is, the LieâPoisson 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 âmagic numberâ ${\boldsymbol{b}}({{\mathfrak{g}}})$ 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 GelfandâTsetlin chains of subalgebras. The 2nd one is known as the âargument shift methodâ of MishchenkoâFomenko. We generalise the GelfandâTsetlin 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.
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