Poisson commutative subalgebras associated with a Cartan subalgebra

Let \({\mathfrak g}\) $\mathfrak{g}$ be a reductive Lie algebra and \(\mathfrak t\subset \mathfrak g\) $\mathfrak{t}\subset \mathfrak{g}$ a Cartan subalgebra. The \(\mathfrak t\) $\mathfrak{t}$ -stable decomposition \({\mathfrak g}=\mathfrak t\oplus {\mathfrak m}\) $\mathfrak{g}=\mathfrak{t}\oplus \mathfrak{m}$ yields a bi-grading of the symmetric algebra \({\mathcal {S}}({\mathfrak g})\) $\mathcal{S}(\mathfrak{g})$ . The subalgebra \({\mathcal {Z}}_{({\mathfrak g},\mathfrak t)}\) ${\mathcal{Z}}_{(\mathfrak{g},\mathfrak{t})}$ generated by the bi-homogenous components of the symmetric invariants \(F\in {\mathcal {S}}({\mathfrak g})^{\mathfrak g}\) $F\in \mathcal{S}{(\mathfrak{g})}^{\mathfrak{g}}$ is known to be Poisson commutative. Furthermore the algebra \({\tilde{{\mathcal {Z}}}}=\textsf{alg}\langle {\mathcal {Z}}_{({\mathfrak g},{\mathfrak t})},{\mathfrak t}\rangle \) $\tilde{\mathcal{Z}}=\mathsf{alg}⟨{\mathcal{Z}}_{(\mathfrak{g},\mathfrak{t})},\mathfrak{t}⟩$ is also Poisson commutative. We investigate relations between \({\tilde{{\mathcal {Z}}}}\) $\tilde{\mathcal{Z}}$ and Mishchenko–Fomenko subalgebras. In type A, we construct a quantisation of \({\tilde{{\mathcal {Z}}}}\) $\tilde{\mathcal{Z}}$ making use of quantum Mishchenko–Fomenko algebras.