The Weight Criterion, Dong Property, and Manin Products of Operads
P. Kolesnikov, B. Sartayev
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.
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