Pre-Leibniz algebras

TL;DR

This paper introduces the cohomology theory of pre-Leibniz algebras, linking it to Maurer-Cartan elements, and explores their deformations and homotopy structures, advancing the algebraic understanding of these objects.

Contribution

It constructs a graded Lie algebra framework for pre-Leibniz algebras and defines their cohomology, providing new tools for studying their deformations and homotopy theory.

Findings

Cohomology of pre-Leibniz algebras is developed and shown to split Leibniz cohomology.

Maurer-Cartan elements characterize pre-Leibniz algebra structures.

Classification of certain homotopy pre-Leibniz algebras is achieved.

Abstract

The notion of pre-Leibniz algebras was recently introduced in the study of Rota-Baxter operators on Leibniz algebras. In this paper, we first construct a graded Lie algebra whose Maurer-Cartan elements are pre-Leibniz algebras. Using this characterization, we define the cohomology of a pre-Leibniz algebra with coefficients in a representation. This cohomology is shown to split the Loday-Pirashvili cohomology of Leibniz algebras. As applications of our cohomology, we study formal and finite order deformations of a pre-Leibniz algebra. Finally, we define homotopy pre-Leibniz algebras and classify some special types of homotopy pre-Leibniz algebras.