Two-sided ideal structure for the Witt and Virasoro enveloping algebras

Determine whether the structural descriptions of two-sided ideals established for the enveloping algebra \(\mathcal U(\mathfrak g[s,s^{-1}])\) and the affine enveloping algebra \(\mathcal U(\widehat{\mathfrak g})\) have corresponding analogues for the enveloping algebras \(\mathcal U(W)\) of the Witt algebra and \(\mathcal U(V)\) of the Virasoro algebra, where the analogous statements are currently conjectural.

Background

The paper compares loop and affine Lie algebras with the Witt algebra WW and its nontrivial central extension, the Virasoro algebra VV. For the symmetric algebras of WW and VV, substantial descriptions of Poisson ideals are known, whereas the corresponding two-sided ideal questions for their enveloping algebras remain unresolved.

The authors identify the missing enveloping-algebra analogues as conjectural. Resolving them would extend the structural theory developed for loop-algebra enveloping algebras to these related infinite-dimensional Lie algebras.

References

For $W$ and $V$, one can almost completely describe Poisson ideals of $\Sa(V)$ and $\Sa(W)$, but the analogous statements for $\Ua(V)$ and $\Ua(W)$ remain conjectural as of this writing (see Section~7 of ).

— Ideals in Enveloping Algebras of Loop Algebras  (2610.22040 - Petukhov et al., 15 Oct 2026) in Section 1, Introduction