The Weight Criterion, Dong Property, and Manin Products of Operads
Abstract
We establish that, for a binary operad Var, the Hadamard product of Var and the operad Nov of Novikov algebras is a binary operad (i.e., coincides with their Manin white product Nov ∘ Var) if and only if the quadratic part of Var governs a 2-variety. The condition Nov ∘ Var = Nov ⊗ Var is known to be equivalent to the existence of an easily verifiable characterization of those differential Var-polynomials which lie in the free derived Var-algebra. For a binary quadratic operad, these conditions are also equivalent to the Dong property of formal distributions. We distinguish the white product defined as a generated sub-operad from its quadratic approximation and give an example in which the canonical epimorphism between them is not injective.
Related
- Anti-successors and anti-replicators, Manin products and Koszul duality
- The Bernoulli property for PSL2ℝ skew products over exponentially mixing base
- Bernoulli property for skew products of exponentially mixing systems over Anosov bases
- The Pairing-Hamiltonian property in Cartesian products of graphs
- AWE: Adaptive Weight Encoding for Exact Integer Matrix Products with Fewer GEMMs on FP4 Tensor Cores
BibTeX
@misc{kolesnikov2026weight,
title = {{The Weight Criterion, Dong Property, and Manin Products of Operads}},
author = {P. Kolesnikov and B. Sartayev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01798},
eprint = {2610.01798},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01798},
}
RIS
TY - UNPB
TI - The Weight Criterion, Dong Property, and Manin Products of Operads
AU - P. Kolesnikov
AU - B. Sartayev
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01798
UR - https://arxiv.org/abs/2610.01798
AB - We establish that, for a binary operad Var, the Hadamard product of Var and the operad Nov of Novikov algebras is a binary operad (i.e., coincides with their Manin white product Nov ∘ Var) if and only if the quadratic part of Var governs a 2-variety. The condition Nov ∘ Var = Nov ⊗ Var is known to be equivalent to the existence of an easily verifiable characterization of those differential Var-polynomials which lie in the free derived Var-algebra. For a binary quadratic operad, these conditions are also equivalent to the Dong property of formal distributions. We distinguish the white product defined as a generated sub-operad from its quadratic approximation and give an example in which the canonical epimorphism between them is not injective.
ER -