Srdjan Vukmirović

Four-dimensional Lie algebras with a para-hypercomplex structure

In this paper we classify 4-dimensional real Lie algebras g admitting an integrable, left invariant, para-hypercomplex structure (Cliff(1,1)-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 g. 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.

Solutions of Yang–Mills equations on generalized Hopf bundles

Trautman has constructed natural self-dual connections on the Hopf bundles over complex and quaternionic projective spaces CPn and HPn; the associated connections are SU(n+1) and Sp(n+1) invariant. Trautman wondered if these connections could be generalized to the case of the corresponding projective spaces defined by indefinite metrics. In this note, we extend the work of Trautman in two different directions. We first define self-dual connections on the Hopf bundles over the projective spaces CP(p,q) and HP(p,q) which are U(p,q+1) and Sp(p,q+1) invariant. We also define self-dual connections over the Hopf bundles associated with the para-complex and para-quarternionic projective spaces CP(p,q) and HP(p,q). Finally, the topology of these projective spaces is investigated.