On Kac’s Jordan superalgebra

Abstract

The group-scheme of automorphisms of the ten-dimensional exceptional Kac's Jordan superalgebra is shown to be isomorphic to the semidirect product of the direct product of two copies of SL 2 by the constant group scheme C 2 . This is used to revisit, extend, and simplify, known results on the classification of the twisted forms of this superalgebra and of its gradings.

Gradings on semisimple algebras

“…In a recent paper [CDE18], the study of gradings (and the affine group scheme of automorphisms) on the 10-dimensional Kac's Jordan superalgebra was reduced to the study of gradings on the direct sum of two copies of the 'tiny' (3-dimensional) Kaplansky superalgebra. In this case the dimension is small enough so that some 'ad hoc' arguments allow a complete classification.…”