From octonions to composition superalgebras via tensor categories
Quantum Algebra
2025-07-17 v2 Rings and Algebras
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.
Cite
@article{arxiv.2205.06559,
title = {From octonions to composition superalgebras via tensor categories},
author = {Alberto Daza-Garcia and Alberto Elduque and Umut Sayin},
journal= {arXiv preprint arXiv:2205.06559},
year = {2025}
}
Comments
21 pages. In the second version some details have been added and some typos corrected