Solvable Lie algebras are not that hypo

TL;DR

This paper investigates obstructions to hypo structures on Lie algebras, providing classification results for solvable Lie algebras that admit such structures, and introduces new geometric and cohomological criteria.

Contribution

It introduces new obstructions based on cohomology and geometry, and classifies solvable Lie algebras admitting hypo structures.

Findings

Obstructions derived from cohomology groups and geometry.

Necessary conditions for hypo structures with fixed almost-contact forms.

Classification of solvable Lie algebras with hypo structures.

Abstract

We study a type of left-invariant structure on Lie groups, or equivalently on Lie algebras. We introduce obstructions to the existence of a hypo structure, namely the 5-dimensional geometry of hypersurfaces in manifolds with holonomy SU(3). The choice of a splitting g^*=V_1 + V_2, and the vanishing of certain associated cohomology groups, determine a first obstruction. We also construct necessary conditions for the existence of a hypo structure with a fixed almost-contact form. For non-unimodular Lie algebras, we derive an obstruction to the existence of a hypo structure, with no choice involved. We apply these methods to classify solvable Lie algebras that admit a hypo structure.