Generalized Castelnuovo-Mumford Regularity for Representations of the Virasoro Algebra
- 1Department of Mathematics, Seoul Women’s University, Seoul 01797, Republic of Korea
Abstract
We compute the generalized noncommutative Castelnuovo-Mumford regularity of irreducible highest-weight modules over the Virasoro algebra Vir. Working over the graded negative universal enveloping algebra A=U(n−)⊂U(Vir) equipped with the level grading (degL−n=n), we analyze three structural classes of representations: generic Verma modules, irreducible quotients with a single singular generator, and rational minimal models. In particular, for the irreducible quotients L(cp,q,hr,s) in minimal models (1≤r≤p−1, 1≤s≤q−1), we show, using the Feigin–Fuchs structure theorem for Verma modules, that the BGG-type resolution of Feigin–Fuchs and Rocha-Caridi–Wallach restricts to a minimal graded free resolution over A with exactly two free summands in each positive homological degree, and we determine all generator degrees separately for odd and even homological degrees. The degree bounds are Ti=(ipq+ηi)2−(rq−sp)2/(4pq), where ηi=|rq−sp| for even i, and ηi=|pq−rq−sp| for odd i. Consequently, the asymptotic regularity exponent is e=2, and the regularity rate is r=pq/4, giving regAL(cp,q,hr,s)=(2,pq/4). The asymptotic regularity rate depends exclusively on the minimal model parameters (p,q) (and thus on the central charge cp,q), remaining invariant under the choice of highest weight hr,s, whereas the linear correction term does depend on hr,s. For (p,q)=(2,3) the result recovers Goncharova’s theorem on the homology of the Lie algebra of vector fields on the line.
Related
- Castelnuovo-Mumford regularity of skew-symmetric matrix Schubert varieties
- Sumsets regularity versus Castelnuovo-Mumford regularity for monomial curves
- Asymptotic Castelnuovo-Mumford regularity of (co)homology modules involving powers of ideals
- Virasoro extensions of generalized Kac-Moody algebras
- Generalized conformal modules over the Virasoro conformal algebra
BibTeX
@article{lee2026generalized,
title = {{Generalized Castelnuovo-Mumford Regularity for Representations of the Virasoro Algebra}},
author = {Dong-il Lee},
journal = {Axioms},
year = {2026},
month = oct,
doi = {10.3390/axioms15110809},
url = {https://www.mdpi.com/2075-1680/15/11/809},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Generalized Castelnuovo-Mumford Regularity for Representations of the Virasoro Algebra
AU - Dong-il Lee
PY - 2026
DA - 2026/10/30
JO - Axioms
DO - 10.3390/axioms15110809
UR - https://www.mdpi.com/2075-1680/15/11/809
AB - We compute the generalized noncommutative Castelnuovo-Mumford regularity of irreducible highest-weight modules over the Virasoro algebra Vir. Working over the graded negative universal enveloping algebra A=U(n−)⊂U(Vir) equipped with the level grading (degL−n=n), we analyze three structural classes of representations: generic Verma modules, irreducible quotients with a single singular generator, and rational minimal models. In particular, for the irreducible quotients L(cp,q,hr,s) in minimal models (1≤r≤p−1, 1≤s≤q−1), we show, using the Feigin–Fuchs structure theorem for Verma modules, that the BGG-type resolution of Feigin–Fuchs and Rocha-Caridi–Wallach restricts to a minimal graded free resolution over A with exactly two free summands in each positive homological degree, and we determine all generator degrees separately for odd and even homological degrees. The degree bounds are Ti=(ipq+ηi)2−(rq−sp)2/(4pq), where ηi=|rq−sp| for even i, and ηi=|pq−rq−sp| for odd i. Consequently, the asymptotic regularity exponent is e=2, and the regularity rate is r=pq/4, giving regAL(cp,q,hr,s)=(2,pq/4). The asymptotic regularity rate depends exclusively on the minimal model parameters (p,q) (and thus on the central charge cp,q), remaining invariant under the choice of highest weight hr,s, whereas the linear correction term does depend on hr,s. For (p,q)=(2,3) the result recovers Goncharova’s theorem on the homology of the Lie algebra of vector fields on the line.
SN - 2075-1680
ER -