Hypercomplex structures on four-dimensional Lie groups
M. L. Barberis,M. L. Barberis +1 more
- 01 Jan 1997
- Vol. 125, Iss: 4, pp 1043-1054
TL;DR: In this paper, the authors classify invariant hypercomplex structures on a 4-dimensional real Lie group G. The purpose of the purpose of this paper is to classify invariants on the real and complex hyperbolic spaces RH4 and CH2, respectively.
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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 4dimensional simply connected Lie groups which admit invariant hypercomplex structures are the additive group IHE of the quaternions, the multiplicative group H* of nonzero quaternions, the solvable Lie groups acting simply transitively on the real and complex hyperbolic spaces, RH4 and CH2, respectively, and the semidirect product C i C. We show that the spaces CH2 and C x C possess an Rp2 of (inequivalent) invariant hypercomplex structures while the remaining groups have only one, up to equivalence. Finally, the corresponding hyperhermitian 4-manifolds are determined.
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Citations
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References
A note on hyperhermitian four-manifolds
Charles P. Boyer
- 01 Jan 1988
TL;DR: On montre que les seules 4-varietes hyperhermitiennes sont, a une equivalence conforme pres, des tores et des surfaces K3 avec leurs structures hyperkahler standards and certaines surfaces de Hopf conformement plates as mentioned in this paper.
130
Visibility and rank one in homogeneous spaces of ≤0
TL;DR: In this paper, the relationship between the visibility axiom and rank one in homogeneous spaces of non-positive curvature has been studied and a complete classification of simply connected and irreducible spaces of dimension < 4 has been obtained.