ACC for graded Poisson ideals of the symmetric algebras

Prove that the Poisson algebras S(W_{a0-1}), S(W), and S(Vir), equipped with their canonical Poisson structures, satisfy the ascending chain condition on graded Poisson ideals.

Background

The paper derives ACC on two-sided ideals for the orbit-homomorphism images by proving ACC on graded Poisson ideals in their associated graded algebras. This motivates the conjecture that the corresponding symmetric Poisson algebras of the one-sided Witt algebra, full Witt algebra, and Virasoro algebra also satisfy ACC on graded Poisson ideals.

If established, this conjecture could provide a strategy for proving ACC on two-sided ideals for the enveloping algebras U(W_{a0-1}), U(W), and U(Vir), which the paper identifies as unresolved.

References

The Poisson algebras $\Sa(\W{-1})$, $\Sa(W)$ and $\Sa(\Vir)$ satisfy ACC on graded Poisson ideals.

— Ideals of homomorphic images of the enveloping algebra of the Witt algebra  (2610.01913 - Pham, 1 Oct 2026) in Conjecture immediately following the discussion of the unresolved ACC problem in Section 2