On seven dimensional quaternionic contact solvable Lie groups

TL;DR

This paper proves the existence of a specific seven-dimensional quaternionic contact manifold with certain geometric properties and classifies seven-dimensional solvable Lie groups with such structures, identifying the quaternionic Heisenberg group as unique in the nilpotent case.

Contribution

It answers an open question about the existence of certain quaternionic contact manifolds and provides a classification approach for seven-dimensional solvable Lie groups with these structures.

Findings

Existence of a seven-dimensional quaternionic contact manifold with closed fundamental 4-form and non-vanishing torsion.

Classification framework for seven-dimensional solvable Lie groups with quaternionic contact structures.

The quaternionic Heisenberg group is the only nilpotent Lie group with such a structure.

Abstract

We answer in the affirmative a question posed by Ivanov and Vassilev on the existence of a seven dimensional quaternionic contact manifold with closed fundamental 4-form and non-vanishing torsion endomorphism. Moreover, we show an approach to the classification of seven dimensional solvable Lie groups having an integrable left invariant quaternionic contact structure. In particular, we prove that the unique seven dimensional nilpotent Lie group admitting such a structure is the quaternionic Heisenberg group.