A rigid Leibniz algebra with non-trivial HL^2

Bakhrom Omirov , Friedrich Wagemann (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

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.

Domaines: Topologie algébrique [math.AT]

Bakhrom Omirov, Friedrich Wagemann. A rigid Leibniz algebra with non-trivial HL^2. 2019. ⟨hal-02030640⟩