Flat hypercomplex nilmanifolds are H-solvable

Abstract

We say that a hypercomplex nilpotent Lie algebra is $\mathbb{H}$-solvable if there exists a sequence of $\mathbb{H}$-invariant subalgebras $\mathfrak{g}_1^{ \mathbb{H}}\supset\mathfrak{g}_2^{ \mathbb{H}}\supset\cdots\supset\mathfrak{g}_{k-1}^{ \mathbb{H}}\supset\mathfrak{g}_k^{ \mathbb{H}}=0,$ such that $[\mathfrak{g}_i^{ \mathbb{H}},\mathfrak{g}_i^{ \mathbb{H}}]\subset\mathfrak{g}^{ \mathbb{H}}_{i+1}.$ Let $N=\Gamma\backslash G$ be a hypercomplex nilmanifold with flat Obata connection and $\mathfrak{g}=Lie(G)$. We prove that the Lie algebra $\mathfrak{g}$ is $ \mathbb{H}$-solvable.