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Young's lattice, invariant ideals, and weak identities
Abstract
For a d-dimensional vector space V over a field of characteristic zero, we show that GL(V)-stable ideals of the universal enveloping algebra of the free two-step nilpotent Lie algebra n2(V) are in natural correspondence with filters in Young's lattice of integer partitions with at most d rows. The same classification holds for GL(V)-stable Poisson ideals of S(V⊕Λ2(V)) with its natural Poisson bracket. As applications, we determine all weak polynomial identities of the n-th Weyl algebra relative to its standard generating subspace and of the universal enveloping algebra of the (2n+1)-dimensional Heisenberg Lie algebra relative to the Lie algebra itself.
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BibTeX
@misc{dotsenko2026youngs,
title = {{Young's lattice, invariant ideals, and weak identities}},
author = {Vladimir Dotsenko and Artem Lopatin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36638},
eprint = {2610.36638},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36638},
}
RIS
TY - UNPB
TI - Young's lattice, invariant ideals, and weak identities
AU - Vladimir Dotsenko
AU - Artem Lopatin
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36638
UR - https://arxiv.org/abs/2610.36638
AB - For a d-dimensional vector space V over a field of characteristic zero, we show that GL(V)-stable ideals of the universal enveloping algebra of the free two-step nilpotent Lie algebra n_2(V) are in natural correspondence with filters in Young's lattice of integer partitions with at most d rows. The same classification holds for GL(V)-stable Poisson ideals of S(V⊕Λ^2(V)) with its natural Poisson bracket. As applications, we determine all weak polynomial identities of the n-th Weyl algebra relative to its standard generating subspace and of the universal enveloping algebra of the (2n+1)-dimensional Heisenberg Lie algebra relative to the Lie algebra itself.
ER -