Biunital Lie Semiheaps and Lie Triple Systems

Abstract

We examine the differential geometry of Lie semiheaps that admit biunital elements. Open Ol'shanskii semigroups, which appear widely across causal geometry and harmonic analysis, provide a large class of examples. It is shown that the subset of biunits is a smoothly embedded Lie subheap. Moreover, there is a canonical Lie triple algebroid associated with any proper biunital Lie semiheap. That is, there is a vector bundle over the manifold of biunits that comes equipped with a ternary bracket that satisfies the axioms of a Lie triple system together with a compatible `higher' anchor map. Although every Lie triple algebroid associated with a proper biunital Lie semiheap is trivialisable, it is not canonically so, unlike the case of the tangent bundle of a Lie group. As concrete examples, we examine the Lie semiheap of real $n\times n$ matrices, and the conformal causal Lie semiheap associated with the Lie group $\mathrm{SU}(2,2)$, which is a double cover of the conformal group of Minkowski spacetime, and the future-directed light cone.