Four-dimensional lie algebras with a para-hypercomplex structure

Abstract

In this paper we classify four-dimensional real Lie algebras g admitting an integrable, left invariant, para-hypercomplex structure. The equivalence classes of compatible structures are classified. The metric of split signature (2; 2), canonically determined by the para-hypercomplex structure, is very convenient in understanding the structure of a. Moreover, these structures provide many examples of left invariant metrics of anti-self-dual metric of split signature. Conformal geometry and the curvature of the canonical metric on the corresponding Lie groups are also discussed. For example, the holonomy algebras of this canonical metrics are determined. © 2010 Rocky Mountain Mathematics Consortium.