Graded Imbeddings in Finite Dimensional Simple Graded Algebras

TL;DR

This paper investigates the embedding problem of finite-dimensional simple graded algebras over algebraically closed fields with finite abelian groups, using group cohomology to characterize graded identities and embeddings.

Contribution

It provides an affirmative answer to the embedding problem for simple graded algebras under specific conditions, employing cohomological methods and graded identity analysis.

Findings

Characterization of G-graded identities via group cohomology.

Conditions for embedding simple G-graded algebras based on graded identities.

Establishment of a correspondence between graded identities and algebra embeddings.

Abstract

Let $\mathbb{F}$ be a field and $\mathsf{G}$ a group. This work is inspired in the following problem: "{\it given a division (simple) $\mathsf{G}$-graded $\mathbb{F}$-algebra, is there any other division (simple) $\mathsf{G}$-graded $\mathbb{F}$-algebra such that the former can be $\mathsf{G}$-imbedded in the latter?}". In this work, we answer this question affirmatively for $\mathbb{F}$ algebraically closed, $\mathsf{G}$ finite abelian, and associative algebras of finite dimension. To prove this, we apply concepts and properties of Group Cohomology. We show $\mathcal{H}^2(H, \mathbb{F}^*)=\mathsf{res}^\mathsf{G}_H \left(\mathcal{H}^2(\mathsf{G}, \mathbb{F}^*)\right)$, where $H$ is a subgroup of $\mathsf{G}$ and $\mathsf{res}^\mathsf{G}_H$ is the restriction homomorphism. Posteriorly, we prove that, given any $H_1,H_2\leq\mathsf{G}$ and $\sigma_i\in\mathcal{Z}^2(H_i,\mathbb{F}^*)$, $i=1,2$, are equivalent: i) $\mathbb{F}^{\sigma_1}[H_1] \stackrel{\mathsf{G}}{\hookrightarrow} \mathbb{F}^{\sigma_2}[H_2]$; ii) $H_1\leq H_2$ and $[\sigma_1]=[\sigma_2]_{H_1}$; iii) $\mathsf{T}^{\mathsf{G}}(\mathbb{F}^{\sigma_2}[H_2])\subseteq \mathsf{T}^{\mathsf{G}}(\mathbb{F}^{\sigma_1}[H_1])$, where $\mathsf{T}^{\mathsf{G}}(\mathbb{F}^{\sigma_i}[H_i])$ is the $\mathsf{G}$T-ideal of graded identities of $\mathbb{F}^{\sigma_i}[H_i]$. Furthermore, we prove that, given $\mathfrak{A}$ and $\mathfrak{B}$ two finite dimensional simple $\mathsf{G}$-graded $\mathbb{F}$-algebras, if $\mathbb{F}$ is algebraically closed, $\mathsf{char}(\mathbb{F}) = 0$ or $\mathsf{char} (\mathbb{F})$ is coprime with the order of each finite subgroup of $\mathsf{G}$, and any subgroup of $\mathsf{G}$ is normal, then $\mathsf{T}^{\mathsf{G}}(\mathfrak{A})\subseteq \mathsf{T}^{\mathsf{G}}(\mathfrak{B})$ iff $\mathfrak{B} \stackrel{\mathsf{G}}{\hookrightarrow} \mathfrak{A}$.