Hyperkähler torsion structures invariant by nilpotent Lie groups

Abstract

We study HKT structures on nilpotent Lie groups and on associated nilmanifolds. We exhibit three weak HKT structures on ${\mathbb{R}}^8$ 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 $\mathfrak{s}p(n)$ 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.

LateX, 12 pages

References in corpus (2)

• Geometry of Hyper-Kähler Connections with Torsion
• The Geometry of (Super) Conformal Quantum Mechanics

Cited by in corpus (6)

• Locally conformally Kähler structures on unimodular Lie groups
• Hypercomplex almost abelian solvmanifolds
• Lie groups as 3-dimensional almost paracontact almost paracomplex Riemannian manifolds
• Applications of the quaternionic Jordan form to hypercomplex geometry
• Algebraic dimension and complex subvarieties of hypercomplex nilmanifolds
• Hypercomplex structures with Hermitian-Norden metrics on four-dimensional Lie algebras