Poisson commutative subalgebras associated with a Cartan subalgebra

Let g g be a reductive Lie algebra and t⊂ g t ⊂ g a Cartan subalgebra. The t t -stable decomposition g= t⊕ m g = t ⊕ m yields a bi-grading of the symmetric algebra S( g) S ( g ) . The subalgebra Z_( g, t) Z ( g , t ) generated by the bi-homogenous components of the symmetric invariants F∈ S( g)ᵍ F ∈ S ( g ) g is known to be Poisson commutative. Furthermore the algebra Z=alg Z_( g, t), t Z ~ = alg ⟨ Z ( g , t ) , t ⟩ is also Poisson commutative. We investigate relations between Z Z ~ and Mishchenko–Fomenko subalgebras. In type , we construct a quantisation of Z Z ~ making use of quantum Mishchenko–Fomenko algebras.