Quaternionic spinors and horospheres in 4-dimensional hyperbolic geometry
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| Format: | Preprint |
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2024
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| author | Mathews, Daniel V. Varsha |
| author_facet | Mathews, Daniel V. Varsha |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_06572 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quaternionic spinors and horospheres in 4-dimensional hyperbolic geometry Mathews, Daniel V. Varsha Geometric Topology Differential Geometry 53C27 (Primary), 15A66 (Secondary) 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. |
| title | Quaternionic spinors and horospheres in 4-dimensional hyperbolic geometry |
| topic | Geometric Topology Differential Geometry 53C27 (Primary), 15A66 (Secondary) |
| url | https://arxiv.org/abs/2412.06572 |