Abelian hypercomplex 8-dimensional nilmanifolds
An invariant Abelian hypercomplex structure on a nilpotent Lie group \(N\) is a pair of anticommuting complex structures \(J_1\) and \(J_2\) such that for each \(J_i\), the corresponding holomorphic vector field \(\Lambda^{1,0}{\mathfrak n}\) is Abelian. The article recalls the notion of a group of type \(H\), defined via an orthogonal decomposition of its Lie algebra into the center \({\mathfrak z}\) and a complement \({\mathfrak v}\) satisfying \([{\mathfrak v},{\mathfrak v}]\subset{\mathfrak z}\). It classifies the invariant Abelian hypercomplex structures on 8-dimensional nilpotent Lie groups \(N\), by showing that they are essentially trivial extensions of groups of type \(H\). The article also considers the topological and geometrical properties of the corresponding nilmanifolds \(\Gamma\setminus N\). In particular, it studies the cohomology of \(\Gamma\setminus N\) in terms of the Lie algebra cohomology of the groups of type \(H\).
scientific article; zbMATH DE number 1422972