The operads of planar forests are Koszul

TL;DR

This paper proves that certain operads related to planar forests are Koszul, providing a duality framework that advances understanding of algebraic structures on planar rooted trees.

Contribution

It introduces the Koszul duals of two quadratic operads on planar forests and establishes their Koszul property, enhancing algebraic tools for planar rooted trees.

Findings

Operads on planar forests are Koszul.

Explicit description of Koszul duals.

Advances in algebraic understanding of planar rooted trees.

Abstract

We describe the Koszul dual of two quadratic operads on planar forests introduced to study the infinitesimal Hopf algebra of planar rooted trees and prove that these operads are Koszul.