Differential graded Lie algebras and Leibniz algebra cohomology

TL;DR

This paper establishes a connection between Leibniz algebras and differential graded Lie algebras, showing how cohomologies are naturally derived and proving a related conjecture.

Contribution

It introduces functors linking Leibniz algebras to differential graded Lie algebras, providing a new perspective on their cohomology theories and confirming a conjecture.

Findings

Leibniz algebras can be interpreted as differential graded Lie algebras.

The functors naturally produce Leibniz and Chevalley-Eilenberg cohomologies.

A conjecture by Pirashvili is proven.

Abstract

In this note, we interpret Leibniz algebras as differential graded Lie algebras. Namely, we consider two functors from the category of Leibniz algebras to that of differential graded Lie algebras and show that they naturally give rise to the Leibniz cohomology and the Chevalley-Eilenberg cohomology. As an application, we prove a conjecture stated by Pirashvili in arXiv:1904.00121 [math.KT].