Hypercomplex structures on special classes of nilpotent and solvable Lie groups (Q2715728)

A hypercomplex structure on a manifold is a triple of complex structures \((J_1, J_2, J_3)\) such that \(J_3=J_1 J_2=-J_2J_1\). The author describes the invariant abelian hypercomplex structure on a step-two nilpotent Lie algebra, by which is meant that the complex structures also preserve the Lie bracket. It is announced without proof that an injective map \(j\) from \(\mathbb R^m\) to the Lie algebra \({sp}(k)\) (\(m\leq k(2k+1)\)) of the symplectic Lie group \(Sp(k)\) gives rise to a step-two nilpotent Lie algebra \({n}=\mathbb R^m \oplus \mathbb R^{2k}\) with an abelian hypercomplex structure and that each step-two nilpotent Lie algebra with an abelian hypercomplex structure arises in this manner.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00032].