Special identities of nonsymmetric bicommutative operations in associative algebra with an operator
Abstract
Let A be an associative non-unital algebra with a linear operator S satisfying aS(S(b)c)=S(aS(b))c, and define a≺ b=aS(b) and a≻ b=S(a)b. We determine the special identities of these two operations. If Kn is the arity-n kernel of the corresponding operad morphism and Dn is the space of consequences of smaller-arity kernel elements, then for every n≥8 the quotient Kn/Dn has dimension n−2. A basis is given by the classes of Fn−2, ρFn−2, Gn−3, ρGn−3, Jn−4, ρJn−4 and Hp,n−p−1, 4≤ p≤ n−5. The upper bound is proved by a direct reduction of singleton sectors, using pure normalization, rigid boundary contraction and the four-singleton corner. The lower bound is obtained from coefficient functionals and a corner projection. In particular, new independent special identities occur in every arity n≥8, and the operadic kernel is not finitely generated.
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BibTeX
@misc{sartayev2026special,
title = {{Special identities of nonsymmetric bicommutative operations in associative algebra with an operator}},
author = {Bauyrzhan Sartayev and Abdibek Ydyrys},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24555},
eprint = {2610.24555},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24555},
}
RIS
TY - UNPB
TI - Special identities of nonsymmetric bicommutative operations in associative algebra with an operator
AU - Bauyrzhan Sartayev
AU - Abdibek Ydyrys
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24555
UR - https://arxiv.org/abs/2610.24555
AB - Let A be an associative non-unital algebra with a linear operator S satisfying aS(S(b)c)=S(aS(b))c, and define a≺ b=aS(b) and a≻ b=S(a)b. We determine the special identities of these two operations. If K_n is the arity-n kernel of the corresponding operad morphism and D_n is the space of consequences of smaller-arity kernel elements, then for every n≥8 the quotient K_n/D_n has dimension n−2. A basis is given by the classes of F_n−2, ρF_n−2, G_n−3, ρG_n−3, J_n−4, ρJ_n−4 and H_p,n−p−1, 4≤ p≤ n−5. The upper bound is proved by a direct reduction of singleton sectors, using pure normalization, rigid boundary contraction and the four-singleton corner. The lower bound is obtained from coefficient functionals and a corner projection. In particular, new independent special identities occur in every arity n≥8, and the operadic kernel is not finitely generated.
ER -