PreprintCC0arXiv
Dimension subrings of Lie rings
Abstract
We show that, unlike the lower central series, the dimension series of a Lie ring need not stabilize when two consecutive terms coincide; in fact, arbitrarily long finite plateaux occur. We prove that Dn(L)/γn+1(L) is central in L/γn+1(L) for every n≥1, in contrast to the group case. If c≥2 and L is metabelian and nilpotent of class at most c, we prove that D2c−1(L)=0. In particular, our results imply that D5(L)⊆γ4(L) for every Lie ring L.
Related
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BibTeX
@misc{ionin2026dimension,
title = {{Dimension subrings of Lie rings}},
author = {Vasily Ionin and Roman Mikhailov and Tima Petrov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02042},
eprint = {2610.02042},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02042},
note = {CC0},
}
RIS
TY - UNPB
TI - Dimension subrings of Lie rings
AU - Vasily Ionin
AU - Roman Mikhailov
AU - Tima Petrov
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02042
UR - https://arxiv.org/abs/2610.02042
AB - We show that, unlike the lower central series, the dimension series of a Lie ring need not stabilize when two consecutive terms coincide; in fact, arbitrarily long finite plateaux occur. We prove that D_n(L)/γ_n+1(L) is central in L/γ_n+1(L) for every n≥1, in contrast to the group case. If c≥2 and L is metabelian and nilpotent of class at most c, we prove that D_2c−1(L)=0. In particular, our results imply that D_5(L)⊆γ_4(L) for every Lie ring L.
ER -