Fr\"{o}licher-Nijenhuis bracket and derived bracket associated to a nonsymmetric operad with multiplication

TL;DR

This paper constructs two graded Lie algebras from a nonsymmetric operad with multiplication, linking Maurer-Cartan elements to Nijenhuis and Rota-Baxter elements, and explores their applications to Loday-type algebras.

Contribution

It introduces explicit graded Lie algebra structures associated with nonsymmetric operads, connecting algebraic operators to Maurer-Cartan elements.

Findings

Explicit forms of brackets for Nijenhuis and Rota-Baxter operators

Identification of Maurer-Cartan elements with specific algebraic operators

Applications to Loday-type algebras

Abstract

This paper aims to construct two graded Lie algebras associated with a nonsymmetric operad with multiplication. Maurer-Cartan elements of these graded Lie algebras correspond respectively to Nijenhuis elements and Rota-Baxter elements for the given multiplication. Explicit forms of these brackets are given to study Nijenhuis operators and Rota-Baxter operators on some Loday-type algebras.