On generic $G$-graded Azumaya algebras

TL;DR

This paper constructs a universal $G$-graded Azumaya algebra that encapsulates the identities of finite dimensional $G$-graded simple algebras over an algebraically closed field, providing a framework for their classification.

Contribution

It introduces a generic $G$-graded Azumaya algebra with a one-to-one ideal correspondence and characterizes $G$-graded simple algebras admitting a division algebra form.

Findings

Constructed a universal $G$-graded Azumaya algebra $$

Established a correspondence between graded ideals and ring ideals

Characterized algebras admitting a $G$-graded division algebra form

Abstract

Let $F$ be an algebraically closed field of characteristic zero and let $G$ be a finite group. Consider $G$-graded simple algebras $A$ which are finite dimensional and $e$-central over $F$, i.e. $Z(A)_{e} := Z(A)\cap A_{e} = F$. For any such algebra we construct a \textit{generic} $G$-graded algebra $\mathcal{U}$ which is \textit{Azumaya} in the following sense. $(1)$ \textit{$($Correspondence of ideals$)$}: There is one to one correspondence between the $G$-graded ideals of $\mathcal{U}$ and the ideals of the ring $R$, the $e$-center of $\mathcal{U}$. $(2)$ \textit{Artin-Procesi condition}: $\mathcal{U}$ satisfies the $G$-graded identities of $A$ and no nonzero $G$-graded homomorphic image of $\mathcal{U}$ satisfies properly more identities. $(3)$ \textit{Generic}: If $B$ is a $G$-graded algebra over a field then it is a specialization of $\mathcal{U}$ along an ideal $\mathfrak{a} \in spec(Z(\mathcal{U})_{e})$ if and only if it is a $G$-graded form of $A$ over its $e$-center. We apply this to characterize finite dimensional $G$-graded simple algebras over $F$ that admit a $G$-graded division algebra form over their $e$-center.