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.

设\({\mathfrak g}\)为还原李代数，\(\mathfrak t\subset \mathfrak g\)为嘉当子代数。 \ (\mathfrak t\)稳定分解\({\mathfrak g}=\mathfrak t\oplus {\mathfrak m}\)产生对称代数\({\mathcal {S}}( {\mathfrak g})\)。由对称不变量\(F\in {\mathcal {S}}的双齐次分量生成的子代数\({\mathcal {Z}}_{({\mathfrak g},\mathfrak t)}\) ({\mathfrak g})^{\mathfrak g}\)已知是泊松交换的。此外，代数\({\tilde{{\mathcal {Z}}}}=\textsf{alg}\langle {\mathcal {Z}}_{({\mathfrak g},{\mathfrak t})}， {\mathfrak t}\rangle \)也是泊松交换律。我们研究\({\tilde{{\mathcal {Z}}}}\)和 Mishchenko-Fomenko 子代数之间的关系。在类型A中，我们利用量子 Mishchenko-Fomenko 代数构造了\({\tilde{{\mathcal {Z}}}}\)的量化。