Poisson commutative subalgebras associated with a Cartan subalgebra

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