Let W_≥ -1=C[t]partial and W=C[t,t^-1]partial be the Witt Lie algebras. The conjecture is that the enveloping algebras operatornameU(W_≥ -1) and operatornameU(W) satisfy the ascending chain condition on two-sided ideals: for every ascending chain of two-sided ideals I_1⊆ I_2⊆… in either enveloping algebra, there exists N≥ 1 such that I_N=I_N+1=….

ACC conjecture for enveloping algebras of the Witt algebra

The conjecture asks whether two-sided ideals in the enveloping algebra of the Witt algebra satisfy the ascending chain condition. Earlier work treated restricted Witt algebras and related quotients, but did not settle the full case.

Known results

A 2017 paper proves the condition for ideals whose associated graded ideals are radical, with related Poisson-ideal results, but leaves the conjecture open.
A 2019 paper proves the condition for completely prime ideals in $U(W_{+})$ and $U(\operatorname{Vir})$ , and establishes just-infinite Gelfand–Kirillov dimension; it explicitly leaves the two-sided-ideal conjecture open.
A 2013 paper proves that $U(W_{+})$ and $U(W)$ are neither left nor right Noetherian, a distinct property.

October 2026 orbit-image result

Tuan Anh Pham's paper proves ACC on two-sided ideals for orbit homomorphic images of the Witt enveloping algebra, in arbitrary Gelfand–Kirillov dimension, and classifies their prime and primitive spectra. This is a claimed partial advance, not a proof for the full Witt enveloping algebra.

Current status (as of October 2026): ACC for the full Witt enveloping algebra remains open; ACC is claimed only for broad orbit-homomorphic images and selected subclasses.

ACC conjecture for enveloping algebras of the Witt algebra

Primary source

Additional references

Progress summary

Solutions 0