Anti-successors and anti-replicators, Manin products and Koszul duality

@misc{bai2026antisuccessors,
  title = {{Anti-successors and anti-replicators, Manin products and Koszul duality}},
  author = {Chengming Bai and Li Guo and Yunhe Sheng and You Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21365},
  eprint = {2610.21365},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21365},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Anti-successors and anti-replicators, Manin products and Koszul duality
AU  - Chengming Bai
AU  - Li Guo
AU  - Yunhe Sheng
AU  - You Wang
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21365
UR  - https://arxiv.org/abs/2610.21365
AB  - 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→ABSu(opd) and opd→ATSu(opd). For binary quadratic operads, these constructions are described by Manin black products: ABSu(opd)≅antiprelie•opd, ATSu(opd)≅antipostlie•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 ADup̂(opd^!) ≅ABSu(opd)^!, ATrip̂(opd^!) ≅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.
ER  -