Hypercomplex almost abelian solvmanifolds

TL;DR

This paper characterizes almost abelian Lie groups with hypercomplex structures, classifies 8-dimensional cases, and explores conditions for flat hyper-Kähler metrics, revealing their geometric and lattice properties.

Contribution

It provides a complete classification of 8-dimensional hypercomplex almost abelian Lie groups and analyzes their geometric structures and lattice existence.

Findings

Obata connection is always flat for these structures.

Certain Lie groups admit HKT and flat hyper-Kähler metrics.

8-dimensional solvmanifolds are either nilmanifolds or have flat hyper-Kähler structures.

Abstract

We give a characterization of almost abelian Lie groups carrying left invariant hypercomplex structures and we show that the corresponding Obata connection is always flat. We determine when such Lie groups admit HKT metrics and study the corresponding Bismut connection. We obtain the classification of hypercomplex almost abelian Lie groups in dimension 8 and determine which ones admit lattices. We show that the corresponding 8-dimensional solvmanifolds are nilmanifolds or admit a flat hyper-K\"ahler metric. Furthermore, we prove that any 8-dimensional compact flat hyper-K\"ahler manifold is a solvmanifold equipped with an invariant hyper-K\"ahler structure. We also construct almost abelian hypercomplex nilmanifolds and solvmanifolds in higher dimensions.