Compatible, split and family Loday-algebras

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.