Cross products, automorphisms, and gradings

TL;DR

This paper classifies automorphism groups and gradings of multilinear cross products on finite-dimensional vector spaces over fields with characteristic not two, providing a comprehensive understanding of their symmetry structures.

Contribution

It determines the automorphism group schemes of multilinear cross products and fully classifies their gradings by abelian groups, up to isomorphism.

Findings

Automorphism group schemes of multilinear cross products are explicitly determined.

Gradings by abelian groups on these structures are completely classified.

The classification is up to isomorphism and applies to fields of characteristic not two.

Abstract

The affine group schemes of automorphisms of the multilinear r-fold cross products on finite-dimensional vectors spaces over fields of characteristic not two are determined. Gradings by abelian groups on these structures, that correspond to morphisms from diagonalizable group schemes into these group schemes of automorphisms, are completely classified, up to isomorphism.