, Volume 16, Issue 1, pp 51-69
Date: 16 Feb 2011

Solvable Lie algebras are not that hypo

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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 five-dimensional geometry of hypersurfaces in manifolds with holonomy SU(3). The choice of a splitting \( {\mathfrak{g}^*} = {V_1} \oplus {V_2} \) , 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 nonunimodular 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.