Cubic Jordan algebras are not a series

Bruce Westbury

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arXiv · 2606.23555

Cubic Jordan algebras are not a series

Bruce Westbury

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Abstract

The idea of the exceptional series is that the exceptional simple Lie algebras should form a series. Since all four simple Lie algebras in the fourth row of the Freudenthal magic square are exceptional it is natural to ask if the remaining rows form a series. A stronger version of this question is that, for the first two rows (corresponding to the real and complex numbers), there is a category defined by a presentation which is a reasonable candidate for the series. Our main results show that neither of these candidates is a series but each consists of a finite set of points. In each case the series is defined by a parameter and we show that the relations imply that this parameter satisfies a polynomial. These two results were obtained by a computer calculation. Our calculation is supported by a website for inspection, and the calculations are certified by Lean 4.

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Bruce Westbury. 2026-06-22. Cubic Jordan algebras are not a series. https://arxiv.org/abs/2606.23555

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