Year: 2026
Type: preprint
Abstract: We develop an operadic theory of anti-splitting and anti-replicating for binary operads. For a binary operad $\opd$, we define its anti-bisuccessor $\ABSu(\opd)$ and anti-trisuccessor $\ATSu(\opd)$. Each generating operation is recovered as the negative sum of two or three split operations, yielding canonical operad morphisms $\opd\to\ABSu(\opd)$ and $\opd\to\ATSu(\opd)$. For binary quadratic operads, these constructions are described by Manin black products: \[ \ABSu(\opd)\cong\antiprelie\bullet\opd, \qquad \ATSu(\opd)\cong\antipostlie\bullet\opd. \] Dually, we also introduce the canonical quadratic anti-duplicator and anti-triplicator, together with their combinatorial counterparts defined by labeled planar binary trees. We show that they coincide precisely under a full-support condition. As applications, we obtain anti-dupli-com and anti-dupli-Lie algebras. We also display an example in which the two constructions differ. Furthermore, we establish the Koszul duality relations \[ \widehat{\ADup}(\opd^!) \cong\ABSu(\opd)^!, \qquad \widehat{\ATrip}(\opd^!) \cong\ATSu(\opd)^!, \] providing an effective description of the canonical quadratic anti-duplicator in terms of the Manin white product. The framework recovers known constructions, including anti-dendriform, anti-pre-Lie and anti-pre-Poisson algebras, and produces further examples and Koszul dual pairs.
Source: arXiv (Cornell University)
Authors: Bai, Chengming, Guo, Li, Sheng, Yunhe, Wang, You
Institutions: Rutgers, The State University of New Jersey, Jilin University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1409244446, G896127649
OpenAlex ID: W7222697192