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.
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References
P. Basarab-Horwath, V. Lahno, R. Zhdanov, The structure of Lie algebras and the classification problem for partial differential equations, Acta Appl. Math. 69 (2001), no. 1. 43–94.
S. Chiossi, A. Fino, Conformally parallel G 2 structures on a class of solvmanifolds, Math. Z. 252 (2006), no. 4, 825–848.
S. Chiossi, S. Salamon, The intrinsic torsion of SU(3) and G2 structures, in: Proceedings of the “International Conference Held to Honour the 60th Birthday of A. M. Naveira”, Valencia (Spain), July 8–14, 2001 (O. Gil-Medrano, V. Miquel, eds.), World Scientific, Singapore, 2002, pp. 115–133.
S. Chiossi, A. Swann, G 2-structures with torsion from half-integrable nilmanifolds, J. Geom. Phys. 54 (2005), 262–285.
D. Conti, S. Salamon, Generalized Killing spinors in dimension 5, Trans. Amer. Math. Soc. 359 (2007), no. 11, 5319–5343.
D. Conti, A. Tomassini, Special symplectic six-manifolds Q. J. Math. 58 (2007), no. 3, 297–311.
D. Conti, Half-flat nilmanifolds, Math. Ann. (2010), in press, DOI: 10.1007/s00208-010-0535-1.
L. C. de Andrés, M. Fernández, A. Fino, L. Ugarte, Contact 5-manifolds with SU(2)-structure, Q. J. Math. 60 (2009), no. 4, 429–459.
A. Diatta, Left invariant contact structures on Lie groups, Differential Geom. Appl. 26 (2008), no. 5, 544–552.
N. Hitchin, Stable forms and special metrics, in: Global Differential Geometry: The Mathematical Legacy of Alfred Gray, Proceedings of International Congress on Differential Geometry, Bilbao (Spain), September 18–23, 2000 (M. Fernández, J.A. Wolf, eds.), Contemporary Mathematics, Vol. 288, Amer. Math. Soc., Providence, RI, 2001, pp. 70–89.
Г. М. Мубаракзянов, Классификация вещесmвенных сmрукmур алгебр Ли пяmого порядка, Изв. вузов, Матем. 34 (1963), но. 3, 99–106. [G. M. Mubarakzyanov, Classification of real structures of Lie algebras of fifth order, Izv. Vyssh. Uchebn. Zaved. Mat. 34 (1963), no. 3, 99–105. (Russian)], Zbl 0166.04201.
F. Schulte-Hengesbach, Half-flat structures on products of three-dimensional Lie groups, J. Geom. Phys. 60 (2010), 1726–1740.
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Conti, D., Fernández, M. & Santisteban, J.A. Solvable Lie algebras are not that hypo. Transformation Groups 16, 51–69 (2011). https://doi.org/10.1007/s00031-011-9127-8
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DOI: https://doi.org/10.1007/s00031-011-9127-8