White Manin product and Hadamard product

TL;DR

The work shows that Alt, Assos, and BiCom operads satisfy a weight criterion that makes their Hadamard product with the Novikov operad coincide with their white Manin product, i.e., $\mathrm{Var}\circ \mathrm{Nov}=\mathrm{Var}\otimes \mathrm{Nov}$. It provides explicit constructions and proofs that these varieties yield $\mathrm{DerVar}$-algebras that are special, with explicit identities for $\mathrm{DerAlt}$, $\mathrm{DerAssos}$, and $\mathrm{DerBiCom}$ and corresponding dimension formulas for $\mathrm{DerVar}(n)$. The results give embedding theorems into $\mathrm{Var}^d$-algebras and supply precise presentations of the derivation-enabled operads, advancing the understanding of special algebras in the Novikov-related operadic framework. These contributions have potential impact on the structure theory of algebras with derivations and on the interplay between Hadamard and white Manin products in operad theory.

Abstract

In this paper, we consider three types of operads: alternative, assosymmetric, and bicommutative. We prove that the Hadamard product of these operads with the Novikov operad coincides with their white Manin product. As an application, we identify a variety of algebras in which all algebras are special.

Theorems & Definitions (22)

• Definition 1
• Lemma 1
• Theorem 1
• Theorem 2
• proof
• Corollary 1
• Theorem 3
• proof
• Lemma 2
• proof