Derived bracket construction up to homotopy and Schroder numbers

TL;DR

This paper introduces a higher derived bracket construction within operads, demonstrating its equivalence to the cobar construction of Leibniz operad, supported by computations involving Schroder numbers.

Contribution

It establishes a novel connection between higher derived brackets and operad theory, specifically linking Lie and Leibniz operads through the cobar construction.

Findings

Higher derived bracket construction of Lie operad is equivalent to the cobar construction of Leibniz operad

The proof involves computing Schroder numbers

Provides new insights into operad relationships and homotopy theory

Abstract

We will introduce the notion of higher derived bracket construction in the category of operads and prove that the higher derived bracket construction of Lie operad is equivalent to the cobar construction of Leibniz operad. The theorem is proved by computing Schroder number.