Derived brackets and sh Leibniz algebras

TL;DR

This paper introduces a comprehensive framework for derived brackets, extending to strong homotopy Leibniz algebras through differential deformation, and explores their links with deformation theory.

Contribution

It generalizes derived bracket construction to sh Leibniz algebras via differential deformation, connecting homotopy algebra and deformation theory.

Findings

Derived brackets can be extended to sh Leibniz algebras.

Deformation of the differential induces a map to sh Leibniz algebra classes.

The framework links homotopy algebra, deformation theory, and derived brackets.

Abstract

We develop a general framework for the construction of various derived brackets. We show that suitably deforming the differential of a graded Leibniz algebra extends the derived bracket construction and leads to the notion of strong homotopy (sh) Leibniz algebra. We discuss the connections among homotopy algebra theory, deformation theory and derived brackets. We prove that the derived bracket construction induces a map from suitably defined deformation theory equivalence classes to the isomorphism classes of sh Leibniz algebras.