Graded semisimple algebras are symmetric

TL;DR

This paper proves that finite dimensional graded semisimple algebras are graded symmetric and explores conditions under which the center of a graded division algebra is symmetric, depending on the grading group and field characteristic.

Contribution

It establishes that graded semisimple algebras are graded symmetric and identifies conditions for the symmetry of centers of graded division algebras.

Findings

Finite dimensional graded semisimple algebras are graded symmetric.

The center of a graded division algebra is symmetric if the group order is coprime to the characteristic.

Provides conditions relating group order, characteristic, and symmetry of centers.

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.