One-parameter deformations of the diassociative and dendriform operads

TL;DR

This paper develops one-parameter deformations of the nonsymmetric dendriform and diassociative operads into symmetric operads, combining Livernet and Loday's polarization with dendriform splitting and Koszul duality.

Contribution

It introduces novel one-parameter deformations of dendriform and diassociative operads using polarization, splitting, and Koszul duality techniques.

Findings

Constructed deformations into symmetric operads.

Unified polarization and splitting approach.

Applied Koszul duality for quadratic operads.

Abstract

Livernet and Loday constructed a polarization of the nonsymmetric associative operad A with one operation into a symmetric operad SA with two operations (the Lie bracket and Jordan product), and defined a one-parameter deformation of SA which includes Poisson algebras. We combine this with the dendriform splitting of an associative operation into the sum of two nonassociative operations, and use Koszul duality for quadratic operads, to construct one-parameter deformations of the nonsymmetric dendriform and diassociative operads into the category of symmetric operads.