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The operad of anti-pre-Lie algebras and its Lie elements
Abstract
We initiate a systematic operadic approach to anti-pre-Lie algebras. We study the operad controlling anti-pre-Lie algebras, its relationship to the operad of Novikov algebras and its cyclic structure. We establish that the Lie bracket of an anti-pre-Lie algebra satisfies the standard identity of degree five. We also establish a quadratic maximality property of the anti-pre-Lie operad with respect to this identity. We then show that the anti-pre-Lie operad and its Koszul dual are not Koszul and give a full description of the latter. Finally, we identify the Lie elements in the Koszul dual operad with the operad of metabelian Lie algebras.
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BibTeX
@misc{bai2026operad,
title = {{The operad of anti-pre-Lie algebras and its Lie elements}},
author = {Chengming Bai and Vladimir Dotsenko and Guilai Liu and Hongjie Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15858},
eprint = {2610.15858},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15858},
}
RIS
TY - UNPB
TI - The operad of anti-pre-Lie algebras and its Lie elements
AU - Chengming Bai
AU - Vladimir Dotsenko
AU - Guilai Liu
AU - Hongjie Wang
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15858
UR - https://arxiv.org/abs/2610.15858
AB - We initiate a systematic operadic approach to anti-pre-Lie algebras. We study the operad controlling anti-pre-Lie algebras, its relationship to the operad of Novikov algebras and its cyclic structure. We establish that the Lie bracket of an anti-pre-Lie algebra satisfies the standard identity of degree five. We also establish a quadratic maximality property of the anti-pre-Lie operad with respect to this identity. We then show that the anti-pre-Lie operad and its Koszul dual are not Koszul and give a full description of the latter. Finally, we identify the Lie elements in the Koszul dual operad with the operad of metabelian Lie algebras.
ER -