Cohomology of solvable Leibniz algebras

TL;DR

This paper extends cohomology theory to solvable Leibniz algebras, overcoming limitations of existing spectral sequences, and generalizes key theorems from Lie algebra cohomology to Leibniz algebras.

Contribution

It introduces new methods to compute Leibniz cohomology, generalizes vanishing theorems, and explores cohomology properties of one-dimensional Leibniz algebras.

Findings

Generalized vanishing theorems for Leibniz algebras

Computed cohomology of one-dimensional Leibniz algebra showing periodicity

Proved Leibniz analogue of a non-vanishing theorem of Dixmier

Abstract

This paper is a sequel to our article [Feldvoss-Wagemann], where we mainly considered semi-simple Leibniz algebras. It turns out that the analogue of the Hochschild-Serre spectral sequence for Leibniz cohomology cannot be applied to many ideals, and therefore this spectral sequence seems not to be applicable for computing the cohomology of non-semi-simple Leibniz algebras. The main idea of the present paper is to use similar tools as developed by Farnsteiner for Hochschild cohomology to work around this. Unfortunately, it does not seem to be possible to relate the cohomology of a Leibniz algebra directly to Hochschild cohomology as is the case for Lie algebras, but all the desired results can be obtained in a similar way. In particular, this enables us to generalize the vanishing theorems of Dixmier and Barnes for nilpotent and (super)solvable Lie algebras to Leibniz algebras. Moreover, we compute the cohomology of the one-dimensional Lie algebra with values in an arbitrary Leibniz bimodule and show that it is periodic with period two. As a consequence, we prove the Leibniz analogue of a non-vanishing theorem of Dixmier. Although not needed in full for the aforementioned results, we prove a Fitting lemma for Leibniz bimodules that might be useful elsewhere.