Gradings on algebras over algebraically closed fields

TL;DR

This paper establishes an equivalence in classifying gradings on certain finite-dimensional nonassociative algebras over algebraically closed fields, linking the problem to scalar extensions.

Contribution

It shows that classifying gradings on these algebras over an algebraically closed field is equivalent to classifying gradings after scalar extension, simplifying the problem.

Findings

Classification of gradings is preserved under scalar extension.

Automorphism group scheme smoothness is key to the equivalence.

Results apply to both isomorphism and equivalence classifications.

Abstract

The classification, both up to isomorphism or up to equivalence, of the gradings on a finite dimensional nonassociative algebra A over an algebraically closed field F, such that its group scheme of automorphisms is smooth, is shown to be equivalent to the corresponding problem for the scalar extension A_K for any algebraically closed field extension K.