Hypercomplex structures on a class of solvable Lie groups

M. L. BARBERIS, I. DOTTI

QUARTERLY JOURNAL OF MATHEMATICS

OXFORD UNIV PRESS

Lugar: Oxford; Año: 1996 vol. 47 p. 389 - 404

0033-5606

This paper concerns the construction of non compact homogeneous manifolds carrying a hypercomplex structure. The case of compact homogeneous hypercomplex manifolds was considered in [Joyce, J. Differential Geom. 1992]. In this article we will focus on the construction of such hypercomplex structures on a class of three step solvable groups. As a first main result we prove that all solvable Lie groups S corresponding to the rank one symmetric spaces of non-compact type do admit S-invariant hypercomplex structures which are compatible with the symmetric metric, except when S corresponds to CH2n, the complex hyperbolic space. In this case the associated complex sphere contains only two points, ±J1, such that the symmetric metric is hermitian and not Kähler. As a second main topic we study in detail the Lie groups G, with dim g?= 2; (g? denotes the derived subalgebra [g,g]) admitting hypercomplex structures. We classify them when dim g? = 1. In particular we obtain examples of solvable Lie groups G such that G x Rs never admits a hypercomplex structure. For dim g? = 2 we prove that if G admits an invariant hypercomplex structure then either G is two step nilpotent or G is locally isomorphic to Rs x (C x C*). The analysis of the cases dim g? = 1 (resp. dim g? = 2) show that the nilmanifolds obtained from R3 x H2n ( resp. R2 x Hn(C)), where H2n (resp. Hn(C)) is the real (resp. complex ) Heisenberg group, have hypercomplex structures. On the other hand, they do not admit Kähler structures by results of Benson-Gordon [Topology 1988]. Some of the results in this paper are contained in the thesis of the first author written under the guidance of the second.