# On Leibniz cohomology

## TL;DR

This paper extends classical Lie algebra cohomology results to Leibniz algebras, proving a Leibniz version of Whitehead's vanishing theorem and computing specific cohomologies using a novel spectral sequence.

## Contribution

It introduces a Leibniz analogue of the Hochschild-Serre spectral sequence and proves new vanishing theorems for Leibniz algebra cohomology.

## Key findings

- Proved Leibniz analogue of Whitehead's vanishing theorem.
- Established the second Whitehead lemma for Leibniz algebras.
- Computed cohomology for several Leibniz algebras with various coefficients.

## Abstract

In this paper we prove the Leibniz analogue of Whitehead's vanishing theorem for the Chevalley-Eilenberg cohomology of Lie algebras. As a consequence, we obtain the second Whitehead lemma for Leibniz algebras. Moreover, we compute the cohomology of several Leibniz algebras with adjoint or irreducible coefficients. Our main tool is a Leibniz analogue of the Hochschild-Serre spectral sequence, which is an extension of (the dual of) a spectral sequence of Pirashvili for Leibniz homology from symmetric bimodules to arbitrary bimodules.