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The exceptional Jordan algebra (2020)
In the early 1930s, Pascual Jordan attempted to formalise the algebraic properties of Hermitian matrices. In particular: Hermitian matrices form a real vector space: we can add and subtract Hermiti…
The subgroup fixing any one of these (without loss of generality, the identity matrix) is the compact real Lie group F4, which also preserves the Jordan product. Possibly the most surprising discovery in this paper is that whilst E6 acts transitively on the positive-definite matrices of determinant 1, this no longer holds when you ‘discretise’! Specifically, as shown in the Elkies-Gross paper, triples of Cayley integers with the norm form an isometric copy of the Leech lattice!
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