Compatible, split and family Loday-algebras

TL;DR

This paper introduces new operads to study compatible, split, and family Loday-algebras, revealing new structures and applications in cohomology and algebra generalizations.

Contribution

It constructs novel operads for compatible and split Loday-algebras, and applies them to cohomology, dendriform, and family Loday-algebras, extending the theoretical framework.

Findings

Cohomology of compatible associative algebra has Gerstenhaber structure

Operad $ ext{Dend}$ generalizes to other Loday-algebras

Introduction of dendriform-family algebras and their homotopy versions

Abstract

Given a nonsymmetric operad $\mathcal{O}$, we first construct two new nonsymmetric operads $\mathcal{O}^{\mathrm{comp}}$ and $\mathcal{O}^{\mathrm{Dend}}$. These operads are respectively useful to study compatible and split Loday-algebras. As an application of the operad $\mathcal{O}^{\mathrm{comp}}$, we show that the cohomology of a compatible associative algebra carries a Gerstenhaber structure. We give an application of the operad $\mathcal{O}^\mathrm{Dend}$ to dendriform algebras and find generalizations to other Loday-algebras. In the end, we construct another operad $\mathrm{Fam}(\mathcal{O}^\Omega)^\mathrm{Dend}$ to study dendriform-family algebras recently introduced in the literature. We also define and study homotopy dendriform-family algebras.