Solvable Lie algebras are not that hypo

Details Solvable Lie algebras are not that hypo Solvable Lie algebras are not that hypo

2010-02-10

arxiv.org/abs/1002.2177v2

Transformation Groups, 16(1):51-69, 2011

Mathematics

Differential Geometry

21 pages; v2: presentation improved, typos corrected, notational conflicts eliminated. To appear in Transformation Groups

Scientific paper

10.1007/s00031-011-9127-8

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.

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