A rigid Leibniz algebra with non-trivial HL^2

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 $M_k⋊\mathfrak{g}$ 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 ${\mathfrak g}={\mathfrak s}{\mathfrak l}_2({\mathbb C})$,wetake$M_k$(resp\mathrm{.}$I_l$) 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 $H{L}^2$ 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 $HL^2({\mathfrak g},{\mathfrak g})=0$.