Gradings on semisimple algebras

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.2010 Mathematics Subject Classification. Primary 17B70, 16W50.

Gradings on tensor products of composition algebras and on the Smirnov algebra

“…Then we will use an isomorphism of schemes to obtain gradings on the tensor product. In this section we will use some results from [CE18] and assume that the base field is algebraically closed, recall that in this case any finitedimensional simple algebra is automatically central-simple (this is a consequence of [Jac78, Theorem 10.1]).…”