Graded semisimple algebras are symmetric

Abstract

We study graded symmetric algebras, which are the symmetric monoids in the monoidal category of vector spaces graded by a group. We show that a finite dimensional graded semisimple algebra is graded symmetric. The center of a symmetric algebra is not necessarily symmetric, but we prove that the center of a finite dimensional graded division algebra is symmetric, provided that the order of the grading group is not divisible by the characteristic of the base field.

Symmetric algebras in categories of corepresentations and smash products

“…It is easy to see that A is (H, ε)-symmetric, i.e. graded symmetric in the terminology of [6], for example by using the results of [7], where the question whether any graded division algebra is graded symmetric is addressed. On the other hand, A is not (H, u)-symmetric.…”