From the Albert algebra to Kac's ten-dimensional Jordan superalgebra via tensor categories in characteristic 5

Abstract

Kac's ten-dimensional simple Jordan superalgebra over an algebraically closed field of characteristic 5 is obtained from a process of semisimplification, via tensor categories, from the exceptional simple Jordan algebra (or Albert algebra), together with a suitable order 5 automorphism.

This explains McCrimmon's ‘bizarre result’ asserting that, in characteristic 5, Kac's superalgebra is a sort of ‘degree 3 Jordan superalgebra’.

As an outcome, the exceptional simple Lie superalgebra $\mathrm{el}(5;5)$, specific of characteristic 5, is obtained from the simple Lie algebra of type $E_8$ and an order 5 automorphism.

In the process, precise recipes to obtain superalgebras from algebras in ${\mathrm{Rep\hspace{0.17em}C}}_p$ (or $\mathrm{Rep}{\alpha }_p$), $p>2$, are given.