Hypercomplex structures on four-dimensional Lie groups

Abstract

Abstract. The purpose of this paper is to classify invariant hypercomplex structures on a 4-dimensional real Lie group G. It is shown that the 4-dimensional simply connected Lie groups which admit invariant hypercomplex structures are the additive group H of the quaternions, the multiplicative group H * of nonzero quaternions, the solvable Lie groups acting simply transitively on the real and complex hyperbolic spaces, RH 4 and CH 2 , respectively, and the semidirect product C C. We show that the spaces CH 2 and C C possess an RP 2 of (inequivalent) invariant hypercomplex structures while the remaining groups have only one, up to equivalence. Finally, the corresponding hyperhermitian 4-manifolds are determined.

Riemannian L p Averaging on Lie Group of Nonzero Quaternions

“…Note that we assume that the tangent space to any element of H * is identified to the linear vector space R 4 [1], i.e., T q H * ∼ = H ∼ = R 4 . In fact, the Lie algebra g H * of the group H * is isomorphic to g H * ∼ = R ⊕ so(3) [12]. In this framework, the Riemannian distance between two quaternions q 1 and q 2 in (H * , ds H * ) is the length of the shortest geodesic path on the manifold H * between both quaternions and is given by…”

Anti-self-dual metrics on Lie groups

“…Four-dimensional Lie groups with a left-invariant hypercomplex structure were classified by Barberis [4], and the above theorem can therefore be viewed as a generalization of this work. Indeed, [4] asserts that G λ admits a left-invariant hypercomplex structure if and only if λ = 0, and that (G 0 , g 1 ) is hyperhermitian.…”

“…Quadratic components of W. In four dimensions, the space W of Weyl tensors has symmetric square 4) and there exist SO(4)-irreducible 9-dimensional representations V ± for which…”

Lie groups of dimension 4 and almost hypercomplex manifolds with Hermitian-Norden metrics

“…A differentiable manifold M of this type has dimension 4n and it is denoted by (M, H, G), where (H, G) is an almost hypercomplex structure with Hermitian-Norden metrics. More precisely, the almost hypercomplex structure H = (J 1 , J 2 , J 3 ) has the following properties: (1,2,3) and the identity I. The quadruplet G = (g, g 1 , g 2 , g 3 ) consists of a a neutral metric g, associated 2-form g 1 and associated neutral metrics g 2 and g 3 on (M, H) having the properties…”