Hyperkähler torsion structures invariant by nilpotent Lie groups

Details Hyperkähler torsion structures invariant by nilpotent Lie groups Hyperkähler torsion structures invariant by nilpotent Lie groups

2001-12-17

arxiv.org/abs/math/0112166v1

Mathematics

Differential Geometry

LateX, 12 pages

Scientific paper

10.1088/0264-9381/19/3/309

We study HKT structures on nilpotent Lie groups and on associated nilmanifolds. We exhibit three weak HKT structures on $\R^8$ which are homogeneous with respect to extensions of Heisenberg type Lie groups. The corresponding hypercomplex structures are of a special kind, called abelian. We prove that on any 2-step nilpotent Lie group all invariant HKT structures arise from abelian hypercomplex structures. Furthermore, we use a correspondence between abelian hypercomplex structures and subspaces of ${\frak sp}(n)$ to produce continuous families of compact and noncompact of manifolds carrying non isometric HKT structures. Finally, geometrical properties of invariant HKT structures on 2-step nilpotent Lie groups are obtained.

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