Weighted rooted trees and deformations of 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.

References in corpus (1)

• Hopf Algebras, Renormalization and Noncommutative Geometry