From octonions to composition superalgebras via tensor categories

Abstract

The nontrivial unital composition superalgebras, of dimension 3 and 6, which exist only in characteristic 3, are obtained from the split Cayley algebra and its order 3 automorphisms, by means of the process of semisimplification of the symmetric tensor category of representations of the cyclic group of order 3. Connections with the extended Freudenthal Magic Square in characteristic 3, that contains some exceptional Lie superalgebras specific of this characteristic are discussed too. In the process, precise recipes to go from (nonassociative) algebras in this tensor category to the corresponding superalgebras are given.

21 pages. In the second version some details have been added and some typos corrected

References in corpus (1)

• Classification of Finite Dimensional Modular Lie Superalgebras with Indecomposable Cartan Matrix

Cited by in corpus (4)

• From the Albert algebra to Kac's ten-dimensional Jordan superalgebra via tensor categories in characteristic 5
• Semisimplifying Lie algebras of $J$-ternary algebras in characteristic $3$
• Semisimplifying Frank Lie algebras
• Semisimplification of contragredient Lie algebras