Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quaternionic spinors and horospheres in 4-dimensional hyperbolic geometry

Daniel V. Mathews, Varsha

TL;DR

The paper extends spinor–horosphere correspondences from complex to quaternionic settings in hyperbolic 4-space, establishing a smooth, SL2-equivariant bijection between quaternionic spinors, spin multiflags, and spin decorated horospheres. Quaternionic lambda lengths are defined via a pseudo-determinant associated with spinor pairs and satisfy a noncommutative Ptolemy relation, grounded in Gel'fand–Retakh quasi-Plücker theory. The work integrates a detailed upper half space description, decorated horospheres, and spin decorations, connecting spinor data to Minkowski light cones and horospherical geometry. Three core strands—spinor geometry, Clifford/Möbius machinery, and noncommutative determinant theory—are unified to provide a robust framework for quaternionic hyperbolic geometry, with potential for further combinatorial and geometric applications in higher dimensional settings.

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-Plücker 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.

Quaternionic spinors and horospheres in 4-dimensional hyperbolic geometry

TL;DR

The paper extends spinor–horosphere correspondences from complex to quaternionic settings in hyperbolic 4-space, establishing a smooth, SL2-equivariant bijection between quaternionic spinors, spin multiflags, and spin decorated horospheres. Quaternionic lambda lengths are defined via a pseudo-determinant associated with spinor pairs and satisfy a noncommutative Ptolemy relation, grounded in Gel'fand–Retakh quasi-Plücker theory. The work integrates a detailed upper half space description, decorated horospheres, and spin decorations, connecting spinor data to Minkowski light cones and horospherical geometry. Three core strands—spinor geometry, Clifford/Möbius machinery, and noncommutative determinant theory—are unified to provide a robust framework for quaternionic hyperbolic geometry, with potential for further combinatorial and geometric applications in higher dimensional settings.

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-Plücker 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.

Paper Structure

This paper contains 59 sections, 65 theorems, 179 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Overview and main results
  3. Quaternionic spinors and paravectors
  4. Multiflags
  5. Horospheres and decorations
  6. Spin isometries of hyperbolic 4-space and Möbius transformations
  7. Explicit description in the upper half space model
  8. Quaternionic lambda lengths in hyperbolic 4-space
  9. Quasideterminants, Plücker and Ptolemy equations
  10. Structure and approach of this paper
  11. Remarks on notation and terminology
  12. Acknowledgments
  13. Geometry of Clifford algebras, quaternions, and paravectors
  14. Clifford algebras in general
  15. Möbius transformations and hyperbolic isometries
  16. ...and 44 more sections

Key Result

Theorem 1

There is an explicit, smooth, bijective, $SL_2 \$$-equivariant correspondence between the following:

Figures (3)

  • Figure 1: Multiflag and decorated horosphere corresponding to a quaternionic spinor.
  • Figure 2: Decorated horospheres as they appear in the upper half space model, with $i$-decoration fields shown in red and $j$-decoration fields in green. This picture is adapted from figure 2 of Mathews_Spinors_horospheres.
  • Figure 3: Complex distance between horospheres, copied from Mathews_Spinors_horospheres.

Theorems & Definitions (160)

  • Theorem 1: Spinor--multiflag--horosphere correspondence
  • Theorem 2: Lambda lengths are pseudo-determinants
  • Theorem 3: Non-commutative Ptolemy equation
  • Definition 1.2.4
  • Definition 1.4.1
  • Definition 1.5.1
  • Theorem 4
  • Proposition 1.8.2
  • Definition 2.6.1
  • Lemma 2.6.2
  • ...and 150 more