Sierra–Walton Conjecture on Noetherianity of Universal Enveloping Algebras

Determine whether, for every Lie algebra L over an algebraically closed field of characteristic 0, the Noetherian property of the universal enveloping algebra U(L) implies that L is finite-dimensional; equivalently, establish that the universal enveloping algebra U(L) of any infinite-dimensional Lie algebra is not Noetherian.

Background

It is classical that if a Lie algebra L is finite-dimensional, then its universal enveloping algebra U(L) is Noetherian. The converse direction is a longstanding problem that has been highlighted by multiple authors and explicitly formulated as a conjecture by Sierra and Walton.

This paper proves the conjecture for a broad class: infinite-dimensional simple Zn-graded Lie algebras (with finite-dimensional homogeneous components) have non-Noetherian universal enveloping algebras. The general case, without the graded simplicity or structural restrictions, remains encapsulated by the Sierra–Walton conjecture.

References

Whether the converse is true has been asked by many authors, among them [B], J. Dixmier, and V. Latyshev. S. Sierra and C. Walton stated this question as a Conjecture.

— Noetherian enveloping algebras of simple graded Lie algebras  (2405.15235 - Andruskiewitsch et al., 2024) in Section 1 (Introduction)

However, we still do not know whether $\Ua(W), \Ua(\W{-1})$ and $\Ua(\Vir)$ satisfy the ACC on two-sided ideals, and this is the topic of future research.

— Ideals of homomorphic images of the enveloping algebra of the Witt algebra  (2610.01913 - Pham, 1 Oct 2026) in Section 2, immediately after the proof of Theorem 2.1 (the theorem labeled \Cref{theo: Bn Cn ACC})