Weighted rooted trees and deformations of operads

TL;DR

This paper introduces a family of operads based on weighted planar rooted trees, interpolating between known operads like NAP, Brace, and pre-Lie, through a deformation parameter.

Contribution

It defines new operads on planar rooted trees and explores their deformation from non-planar operads, connecting various operadic structures via a scalar parameter.

Findings

Defined operad $ ext{B}^0$ on planar rooted trees.

Introduced a deformation family $ ext{B}^ ext{lambda}$ interpolating between known operads.

Established a map linking non-planar and planar rooted trees, connecting $NAP$, Brace, and pre-Lie operads.

Abstract

We will define an operad $\mathcal{B}^0$ on planar rooted trees. $\mathcal{B}^0$ is analgous to the $NAP$-operad in the non-planar tree setting. We will define a family of "current-preserving" operads $\mathcal{B}^\lambda$ depending on a scalar parameter $\lambda$, which can be seen as a deformation of the operad $\mathcal{B}^0$. Forgetting the extra "current preserving" notion above give back the Brace operad for $\lambda=1$ and the $\mathcal{B}^0$ operad for $\lambda=0$. A natural map from non-planar rooted trees to plane ones gives back the current-preserving interpolation between $NAP$ and pre-Lie investigated in a previous article.