A rigid Leibniz algebra with non-trivial HL²

In this article, we generalize Richardson's example of a rigid Lie algebra with non-trivial H² to the Leibniz setting. Namely, we consider the hemisemidirect product h of a semidirect product Lie algebra M_k⋊g of a simple Lie algebra g with some non-trivial irreducible g-module M_k with a non-trivial irreducible g-module I_l. Then for g=sl₂(C), we take M_k (resp. I_l) to be the standard irreducible sl₂(C)-module of dimension k+1 (resp. l+1). Assume k2>5 is an odd integer and l>2 is odd, then we show that the Leibniz algebra h is geometrically rigid and has non-trivial HL² with adjoint coefficients. We close the article with an appendix where we record further results on the question whether H²(g,g)=0 implies HL²(g,g)=0.