From octonions to composition superalgebras via tensor categories

TL;DR

This paper explores the construction of unique composition superalgebras in characteristic 3 from the Cayley algebra using tensor categories, linking them to exceptional Lie superalgebras and providing explicit construction methods.

Contribution

It introduces a novel approach to derive nontrivial composition superalgebras in characteristic 3 via semisimplification of tensor categories, connecting algebraic structures with the extended Freudenthal Magic Square.

Findings

Construction of 3- and 6-dimensional superalgebras in characteristic 3

Explicit recipes for transitioning from algebras to superalgebras

Connections established between superalgebras and exceptional Lie superalgebras

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.