Transformation Groups

, Volume 16, Issue 1, pp 51–69

Solvable Lie algebras are not that hypo

  • Diego Conti
  • Marisa Fernández
  • José A. Santisteban
Article

DOI: 10.1007/s00031-011-9127-8

Cite this article as:
Conti, D., Fernández, M. & Santisteban, J.A. Transformation Groups (2011) 16: 51. doi:10.1007/s00031-011-9127-8

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 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.

AMS classification

53C25 (primary)53C15, 17B30, 53D15 (secondary)

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Diego Conti
    • 1
  • Marisa Fernández
    • 2
  • José A. Santisteban
    • 2
  1. 1.Dipartimento di Matematica e ApplicazioniUniversità di Milano BicoccaMilanoItaly
  2. 2.Facultad de Ciencia y Tecnología, Departamento de MatemáticasUniversidad del País VascoBilbaoSpain