Odd-rank maximal ideals at collapsing levels of type D
Abstract
We determine the defining ideal of the simple affine vertex algebra L2−ℓ(so2ℓ) for every odd ℓ≥5. Perše's quadratic singular vector alone generates the maximal ideal of the universal affine vertex algebra at this level. Together with the established even-rank presentation, this gives a complete parity-dependent description of this type-D collapsing family at k=2−ℓ: one quadratic generator in odd rank, and a quadratic generator together with two Pfaffian generators in even rank. The proof establishes a rank reduction for the quadratic quotients under minimal Drinfeld--Sokolov reduction, valid in both parities. Nonvanishing of reduction on nonzero graded subquotients then lifts simplicity along the odd-rank chain from the known base case D3≅ A3 at level −1.
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BibTeX
@misc{jin2026oddrank,
title = {{Odd-rank maximal ideals at collapsing levels of type $D$}},
author = {Sihai Jin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12706},
eprint = {2609.12706},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12706},
}
RIS
TY - UNPB
TI - Odd-rank maximal ideals at collapsing levels of type D
AU - Sihai Jin
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12706
UR - https://arxiv.org/abs/2609.12706
AB - We determine the defining ideal of the simple affine vertex algebra L_2−ℓ(so_2ℓ) for every odd ℓ≥5. Perše's quadratic singular vector alone generates the maximal ideal of the universal affine vertex algebra at this level. Together with the established even-rank presentation, this gives a complete parity-dependent description of this type-D collapsing family at k=2−ℓ: one quadratic generator in odd rank, and a quadratic generator together with two Pfaffian generators in even rank. The proof establishes a rank reduction for the quadratic quotients under minimal Drinfeld--Sokolov reduction, valid in both parities. Nonvanishing of reduction on nonzero graded subquotients then lifts simplicity along the odd-rank chain from the known base case D_3≅ A_3 at level −1.
ER -