A rigid Leibniz algebra with non-trivial HL^2

TL;DR

This paper constructs a specific Leibniz algebra that is geometrically rigid yet has a non-trivial second cohomology group, extending Richardson's example from Lie to Leibniz algebras.

Contribution

It generalizes Richardson's example to Leibniz algebras, showing the existence of rigid Leibniz algebras with non-trivial second cohomology.

Findings

Constructed a Leibniz algebra with non-trivial HL^2

Proved the algebra is geometrically rigid under certain conditions

Provided results on the relation between Lie and Leibniz cohomology

Abstract

In this article, we generalize Richardson's example of a rigid Lie algebra with non-trivial $H^2$ to the Leibniz setting. Namely, we consider the hemisemidirect product ${\mathfrak h}$ of a semidirect product Lie algebra $M_k\rtimes{\mathfrak g}$ of a simple Lie algebra ${\mathfrak g}$ with some non-trivial irreducible ${\mathfrak g}$-module $M_k$ with a non-trivial irreducible ${\mathfrak g}$-module $I_l$. Then for ${\mathfrak g}={\mathfrak s}{\mathfrak l}_2({\mathbb C})$, we take $M_k$ (resp. $I_l$) to be the standard irreducible ${\mathfrak s}{\mathfrak l}_2({\mathbb C})$-module of dimension $k+1$ (resp. $l+1$). Assume $\frac{k}{2}>5$ is an odd integer and $l>2$ is odd, then we show that the Leibniz algebra ${\mathfrak h}$ is geometrically rigid and has non-trivial $HL^2$ with adjoint coefficients. We close the article with an appendix where we record further results on the question whether $H^2({\mathfrak g},{\mathfrak g})=0$ implies $HL^2({\mathfrak g},{\mathfrak g})=0$.