A rigid Leibniz algebra with non-trivial HL2

Abstract

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 Mk⋊g of a simple Lie algebra g with some non-trivial irreducible g-module Mk with a non-trivial irreducible g-module Il. Then for g=sl2(C), we take Mk (resp. Il) 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.