Gradings on semisimple algebras

TL;DR

This paper classifies gradings by abelian groups on semisimple nonassociative algebras, reducing the problem to simple algebras using loop algebras and introducing a new approach to group-gradings.

Contribution

It introduces a framework for understanding gradings on semisimple algebras by connecting them to simple algebras via loop algebras and defining free products of group-gradings.

Findings

Reduction of grading classification to simple algebras

Development of a definition for free products of group-gradings

Application of loop algebras in grading classification

Abstract

The classification of gradings by abelian groups on finite direct sums of simple finite-dimensional nonassociative algebras over an algebraically closed field is reduced, by means of the use of loop algebras, to the corresponding problem for simple algebras. This requires a good definition of (free) products of group-gradings.