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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Hypercomplex structures
on four-dimensional Lie groups


Author: María Laura Barberis
Journal: Proc. Amer. Math. Soc. 125 (1997), 1043-1054
MSC (1991): Primary 32M10, 53C25, 53C56; Secondary 32M15, 53C55
MathSciNet review: 1353375
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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 $4$-dimensional simply connected Lie groups which admit invariant hypercomplex structures are the additive group $\mathbb H$ of the quaternions, the multiplicative group ${\mathbb H}^*$ of nonzero quaternions, the solvable Lie groups acting simply transitively on the real and complex hyperbolic spaces, ${\mathbb R}H^4$ and ${\mathbb C}H^2$, respectively, and the semidirect product ${\mathbb C}\rtimes {\mathbb C}$. We show that the spaces ${\mathbb C}H^2$ and ${\mathbb C}\rtimes {\mathbb C}\,$ possess an ${\mathbb R}P^2$ 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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Additional Information

María Laura Barberis
Affiliation: FaMAF, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 - Córdoba, Argentina
Address at time of publication: Department of Mathematics, University of California at San Diego, La Jolla, California 92093
Email: barberis@mate.uncor.edu, mbarberis@ucsd.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03611-3
PII: S 0002-9939(97)03611-3
Keywords: Hypercomplex structure (hcs), hyperhermitian metric
Received by editor(s): March 9, 1994
Received by editor(s) in revised form: September 5, 1995
Additional Notes: The author was partially supported by Conicor and Conicet (Argentina).
Communicated by: Roe Goodman
Article copyright: © Copyright 1997 American Mathematical Society