Gradings on semisimple algebras

Abstract

In this article the classification of gradings by abelian groups on finite direct sums of simple finite-dimensional nonassociative algebras over an algebraically closed field is studied. The problem is reduced to the corresponding classification for simple algebras through the use of loop algebras and suitable definitions of product gradings. Conditions under which semisimple graded algebras can be decomposed into graded-simple components are established, and fine group gradings on semisimple algebras are described and classified up to equivalence.