Abelian hypercomplex 8-dimensional nilmanifolds
type: Item
instance of: scholarly article
title: Abelian hypercomplex 8-dimensional nilmanifolds
zbMATH Open document ID: 0945.53031
DOI: 10.1023/A:1006656824085
author: Anna Fino
author: I. G. Dotti
published in: Annals of Global Analysis and Geometry
publication date: 2000-10-03T00:00:00Z
review text: 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 \(H\).
reviewed by: Meng-Kiat Chuah
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53C56
Mathematics Subject Classification ID: 22E25
zbMATH DE Number: 1422972
zbMATH Keywords: abelian hypercomplex structure
zbMATH Keywords: nilpotent Lie group
zbMATH Keywords: groups of type \(H\)
zbMATH Keywords: nilmanifold
zbMATH Keywords: Lie algebra cohomology
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1422972