Poisson commutative subalgebras associated with a Cartan subalgebra

Abstract

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