A unified approach to the definition of anti-algebras
Abstract
We introduce the Marseille anti-construction Ma for varieties of algebras defined by multilinear identities of degree $3$, obtained by changing the sign of the right-nested part of each defining polynomial. This gives, for any variety Var, a new variety Ma-Var that generalizes antiassociative algebras and the recently studied anti-Novikov and anti-right-symmetric (pre-Jacobi--Jordan) algebras. We describe free objects in several Ma-varieties. On the operadic level, over a field of characteristic char Bbbk≠ 2, we show (Ma−Var)(!)=Ma−(Var(!)) and that the Dong property is preserved by Ma. Finally, we obtain the necessary and sufficient condition for Ma−Var=Var. The second part of the paper is dedicated to another type of anti-algebras, called ℳanin anti-algebras. We give the description of this type of algebras via calculation of the white Manin product of suitable operads and present various examples of known ℳanin anti-algebras. In the end, we proved that the symmetric affinization of an algebra gives a Jacobi-Jordan algebra if and only if the original algebra is a ℳa-Novikov algebra, that is, an analog of a well-known result about affinization of Novikov algebras given by Balinskii and Novikov.
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BibTeX
@misc{kaygorodov2026unified,
title = {{A unified approach to the definition of anti-algebras}},
author = {Ivan Kaygorodov and Abror Khudoyberdiyev and Bauyrzhan Sartayev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12479},
eprint = {2610.12479},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12479},
note = {CC0},
}
RIS
TY - UNPB
TI - A unified approach to the definition of anti-algebras
AU - Ivan Kaygorodov
AU - Abror Khudoyberdiyev
AU - Bauyrzhan Sartayev
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.12479
UR - https://arxiv.org/abs/2610.12479
AB - We introduce the Marseille anti-construction Ma for varieties of algebras defined by multilinear identities of degree $3$, obtained by changing the sign of the right-nested part of each defining polynomial. This gives, for any variety Var, a new variety Ma-Var that generalizes antiassociative algebras and the recently studied anti-Novikov and anti-right-symmetric (pre-Jacobi--Jordan) algebras. We describe free objects in several Ma-varieties. On the operadic level, over a field of characteristic char Bbbk≠ 2, we show (Ma−Var)^(!)=Ma−(Var^(!)) and that the Dong property is preserved by Ma. Finally, we obtain the necessary and sufficient condition for Ma−Var=Var. The second part of the paper is dedicated to another type of anti-algebras, called ℳanin anti-algebras. We give the description of this type of algebras via calculation of the white Manin product of suitable operads and present various examples of known ℳanin anti-algebras. In the end, we proved that the symmetric affinization of an algebra gives a Jacobi-Jordan algebra if and only if the original algebra is a ℳa-Novikov algebra, that is, an analog of a well-known result about affinization of Novikov algebras given by Balinskii and Novikov.
ER -