Quaternionic spinors and horospheres in 4-dimensional hyperbolic geometry

TL;DR

This paper establishes new correspondences between quaternionic spinors, flags in Minkowski space, and horospheres in 4D hyperbolic space, extending classical geometric concepts with quaternionic and non-commutative structures.

Contribution

It introduces a novel framework linking quaternionic spinors, hyperbolic geometry, and non-commutative algebra, generalizing previous lower-dimensional results.

Findings

Quaternionic lambda lengths generalize classical lengths to 4D with quaternionic values.

Lambda lengths satisfy a non-commutative Ptolemy relation.

New geometric and topological structures are identified in the quaternionic hyperbolic setting.

Abstract

We give explicit bijective correspondences between three families of objects: certain pairs of quaternions, which we regard as spinors; certain flags in (1+4)-dimensional Minkowski space; and horospheres in 4-dimensional hyperbolic space decorated with certain pairs of spinorial directions. These correspondences generalise previous work of the first author, Penrose--Rindler, and Penner in lower dimensions, and use the description of 4-dimensional hyperbolic isometries via Clifford matrices studied by Ahlfors and others. We show that lambda lengths generalise to 4 dimensions, where they take quaternionic values, and are given by a certain bilinear form on quaternionic spinors. They satisfy a non-commutative Ptolemy equation, arising from quasi-Pl\"ucker relations in the Gel'fand--Retakh theory of noncommutative determinants. We also study various structures of geometric and topological interest that arise in the process.