Three-Dimensional Geometry in Exceptional Algebra
Abstract
We review some topics in "exceptional mathematics'' from the perspective of 3-dimensional geometry: the octonions π, the split octonions π', the bioctonions πC β β βR π, the complex Albert algebra h3(πC), and the complex form of the exceptional Lie algebra e6. We show how to functorially build an algebra isomorphic to π from any 3d complex vector space equipped with an inner product and complex volume form. Similarly, we build one isomorphic to π' starting from a 3d real vector space equipped with a volume form, and one isomorphic to πC starting from a 3d complex vector space equipped with a complex volume form. We give applications to 3-dimensional real and complex manifolds. Finally, we describe how to build an Jordan algebra isomorphic to h3(πC) starting from three 3d complex vector spaces equipped with complex volume forms. This last construction gives a nice explicit description of the complex Lie algebra e6 and its subalgebra sl(3,β) β sl(3,β) β sl(3,β).
Related
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BibTeX
@misc{baez2026threedimensional,
title = {{Three-Dimensional Geometry in Exceptional Algebra}},
author = {John C. Baez and Susanne Pumpluen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07538},
eprint = {2609.07538},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07538},
}
RIS
TY - UNPB
TI - Three-Dimensional Geometry in Exceptional Algebra
AU - John C. Baez
AU - Susanne Pumpluen
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07538
UR - https://arxiv.org/abs/2609.07538
AB - We review some topics in "exceptional mathematics'' from the perspective of 3-dimensional geometry: the octonions π, the split octonions π', the bioctonions π_C β
β β_R π, the complex Albert algebra h_3(π_C), and the complex form of the exceptional Lie algebra e_6. We show how to functorially build an algebra isomorphic to π from any 3d complex vector space equipped with an inner product and complex volume form. Similarly, we build one isomorphic to π' starting from a 3d real vector space equipped with a volume form, and one isomorphic to π_C starting from a 3d complex vector space equipped with a complex volume form. We give applications to 3-dimensional real and complex manifolds. Finally, we describe how to build an Jordan algebra isomorphic to h_3(π_C) starting from three 3d complex vector spaces equipped with complex volume forms. This last construction gives a nice explicit description of the complex Lie algebra e_6 and its subalgebra sl(3,β) β sl(3,β) β sl(3,β).
ER -