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

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 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 Rep C p (or Rep α p ), p > 2 , are given.

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